1. Venir – to come
Present indicative (presente del indicativo):
yo vengo, tú vienes, usted/él/ella viene
nosotros/as venimos, vosotros/as venís, ustedes/ellos/ellas vienen
Preterite (pretérito): yo vine, tú viniste, usted/él/ella vino
nosotros/as vinimos, vosotros/as vinisteis, ustedes/ellos/ellas vinieron
Imperfect indicative (imperfecto del indicativo): yo venía, tú venías, usted/él/ella venía
nosotros/as veníamos, vosotros/as veníais, ustedes/ellos/ellas venían
Future (futuro): yo vendré, tú vendrás, usted/él/ella vendrá
nosotros/as vendremos, vosotros/as vendréis, ustedes/ellos/ellas vendrán
2. infinitive (infinitivo): empezar - to start , to begin
Gerund (gerundio): empezando
Present indicative (presente del indicativo):
yo empiezo, tú empiezas, usted/él/ella empieza
nosotros/as empezamos, vosotros/as empezáis, ustedes/ellos/ellas empiezan
Preterite (pretérito): yo empecé, tu empezaste, usted/él/ella empezó
nosotros/as empezamos, vosotros/as empezasteis, ustedes/ellos/ellas empezaron
Imperfect indicative (imperfecto del indicativo):
yo empezaba, tú empezabas, usted/él/ella empezaba
nosotros/as empezábamos, vosotros/as empezabais, ustedes/ellos/ellas empezaban
Future (futuro): yo empezaré, tú empezarás, usted/él/ella empezará
nosotros/as empezaremos, vosotros/as empezaréis, ustedes/ellos/ellas empezarán
3. Infinitive (infinitivo): traer – to bring
Gerund (gerundio): trayendo
Participle (participio): traído
Present indicative (presente del indicativo):
yo traigo, tú traes, usted/él/ella trae
nosotros/as traemos, vosotros/as traéis, ustedes/ellos/ellas traen
Preterite (pretérito): yo traje, tu trajiste, usted/él/ella trajo
nosotros/as trajimos, vosotros/as trajisteis, ustedes/ellos/ellas trajeron
Imperfect indicative (imperfecto del indicativo):
yo traía, tú traías, usted/él/ella traía
nosotros/as traíamos, vosotros/as traíais, ustedes/ellos/ellas traían
Future (futuro): yo traeré, tú traerás, usted/él/ella traerá
nosotros/as traeremos, vosotros/as traeréis, ustedes/ellos/ellas traerán
4. Infinitive (infinitivo): querer - to want
Gerund (gerundio): queriendo (wanting)
Participle (participio): querido (wanted)
Present indicative (presente del indicativo):
yo quiero, tú quieres, usted/él/ella quiere
nosotros/as queremos, vosotros/as queréis, ustedes/ellos/ellas quieren
(I want, you want, etc.)
Preterite (pretérito): yo quise, tu quisiste, usted/él/ella quiso
nosotros/as quisimos, vosotros/as quisisteis, ustedes/ellos/ellas quisieron
(I wanted, you wanted, etc.)
Imperfect indicative (imperfecto del indicativo):
yo quería, tú querías, usted/él/ella quería
nosotros/as queríamos, vosotros/as queríais, ustedes/ellos/ellas querían
(I used to want, you used to want, etc.)
Future (futuro): yo querré, tú querrás, usted/él/ella querrá
nosotros/as querremos, vosotros/as querréis, ustedes/ellos/ellas querrán
(I will want, you will want, etc.)
5. Infinitive (infinitivo): ver (to see)
Gerund (gerundio): viendo (seeing)
Participle (participio): visto (seen)
Present indicative (presente del indicativo):
yo veo, tú ves, usted/él/ella ve
nosotros/as vemos, vosotros/as véis, ustedes/ellos/ellas ven
(I see, you see, etc.)
Preterite (pretérito): yo vi, tu viste, usted/él/ella vio
nosotros/as vimos, vosotros/as visteis, ustedes/ellos/ellas vieron
(I saw, you saw, etc.)
Imperfect indicativo (imperfecto del indicativo):
yo veía, tú veías, usted/él/ella veía
nosotros/as veíamos, vosotros/as veíais, ustedes/ellos/ellas veían
(I used to see, you used to see, etc.)
Future (futuro): yo veré, tú verás, usted/él/ella verá
nosotros/as veremos, vosotros/as veréis, ustedes/ellos/ellas verán
(I will see, you will see, etc.)
6. Infinitive (infinitivo) to go: ir
Following are the conjugations for the verb ir, which typically means "to go." Irregular forms are indicated in boldface. Other translations other than those given are possible.
Gerund (gerundio) going: yendo
Participle (participio)gone: ido
Present indicative (presente del indicativo) I go, you go, etc.:
yo voy, tú vas, usted/él/ella va
nosotros/as vamos, vosotros/as vais, ustedes/ellos/ellas van
Preterite (pretérito) I went, you went, etc.: yo fui, tu fuiste, usted/él/ella fue
nosotros/as fuimos, vosotros/as fuisteis, ustedes/ellos/ellas fueron
Imperfect indicative (imperfecto del indicativo) I used to go, you used to go, etc.: yo iba, tú ibas, usted/él/ella iba
nosotros/as íbamos, vosotros/as ibais, ustedes/ellos/ellas iban
Future (futuro) I will go, you will go, etc.:
yo iré, tú irás, usted/él/ella irá
nosotros/as iremos, vosotros/as iréis, ustedes/ellos/ellas irán
Note: The preterite conjugation is the same as that of the verb ser. The context typically will indicate which verb is being conjugated.
7. Infinitive (infinitivo) haber – to have
Haber is not used in the imperative mood.
Note: Haber is used most frequently as an auxiliary verb to form the perfect tenses. It also has two other uses: In the third-person present indicative form of hay, it can mean "there is" or "there are." It is also used in the phrase hay que, meaning "it is necessary." With those meanings, haber also can be used in the other third-person tenses.
Gerund (gerundio) habiendo
Past participle (participio) habido
Present indicative (presente del indicativo)
yo he, tú has, usted/él/ella ha (hay)
nosotros/as hemos, vosotros/as habéis, ustedes/ellos/ellas han (hay)
Preterite (pretérito)
yo hube, tu hubiste, usted/él/ella hubo
nosotros/as hubimos, vosotros/as hubisteis, ustedes/ellos/ellas hubieron
Imperfect indicativo (imperfecto del indicativo)
yo había, tú habías, usted/él/ella había
nosotros/as habíamos, vosotros/as habíais, ustedes/ellos/ellas habían
Future (futuro): yo habré, tú habrás, usted/él/ella habrá
nosotros/as habremos, vosotros/as habréis, ustedes/ellos/ellas habrán
8. Poder – to be able
Gerund (gerundio): pudiendo
Participle (participio): podido
Present indicative:
yo puedo, tú puedes, usted/él/ella puede
nosotros/as podemos, vosotros/as podéis, ustedes/ellos/ellas pueden
Preterite (pretérito):
yo pude, tu pudiste, usted/él/ella pudo
nosotros/as pudimos, vosotros/as pudisteis, ustedes/ellos/ellas pudieron
Imperfect indicativo (imperfecto del indicativo):
yo podía, tú podías, usted/él/ella podía
nosotros/as podíamos, vosotros/as podíais, ustedes/ellos/ellas podían
Future (futuro):
yo podré, tú podrás, usted/él/ella podrá
nosotros/as podremos, vosotros/as podréis, ustedes/ellos/ellas podrán
9. Conjugation of almorzar – to eat lunch
Infinitive (infinitivo) Almorzar
Gerund (gerundio) Almorzando
Participle (participio) Almorzado
Present indicative(presente del inidicativo)
yo almuerzo, tú almuerzas, usted/él/ella almuerza
nosotros/as almorzamos, vosotros/as almorzáis, ustedes/ellos/ellas almuerzan
Preterite (pretérito)yo almorcé, tu almorzaste, usted/él/ella almorzó
nosotros/as almorzamos, vosotros/as almorzasteis, ustedes/ellos/ellas almorzaron
Imperfect indicativo
(imperfecto del indicativo) yo almorzaba, tú almorzabas, usted/él/ella almorzaba nosotros/as almorzábamos, vosotros/as almorzabais, ustedes/ellos/ellas almorzaban
Future (futuro) yo almorzaré, tú almorzarás, usted/él/ella almorzará
nosotros/as almorzaremos, vosotros/as almorzaréis, ustedes/ellos/ellas almorzarán
10. Conjugation of andar – to walk
Infinitivo - andar
Gerund (gerundio)- andando
Participle (participio) - andado
Present indicative (presente del inidicativo)
yo ando, tú andas, usted/él/ella anda
nosotros/as andamos, vosotros/as andáis, ustedes/ellos/ellas andan
Preterite (pretérito) yo anduve, tu anduviste, usted/él/ella anduvo, nosotros/as anduvimos, vosotros/as anduvisteis, ustedes/ellos/ellas anduvieron
Imperfect indicativo (imperfecto del indicativo)
yo andaba, tú andabas, usted/él/ella andaba
nosotros/as andábamos, vosotros/as andabais, ustedes/ellos/ellas andaban
Future (futuro) yo andaré, tú andarás, usted/él/ella andará
nosotros/as andaremos, vosotros/as andaréis, ustedes/ellos/ellas andarán
11. Conjugation of argüir - to deduce, prove, argue
Infinitive (infinitivo) argüir
Gerund (gerundio) arguyendo
Participle (participio) argüido
Presente yo arguyo, tú arguyes, usted/él/ella arguye
nosotros/as argüimos, vosotros/as argüís, ustedes/ellos/ellas arguyeren
Preterite (pretérito)yo argüí, tu argüiste, usted/él/ella arguyó
nosotros/as argüimos, vosotros/as argüisteis, ustedes/ellos/ellas argüieron
Imperfect indicative
(imperfecto del indicativo) yo arguya, tú arguyas, usted/él/ella arguya
nosotros/as arguyamos, vosotros/as arguyais, ustedes/ellos/ellas arguyan
Future (futuro) yo argüiré, tú argüirás, usted/él/ella argüirá
nosotros/as argüiremos, vosotros/as argüiréis, ustedes/ellos/ellas argüirán
12. Conjugation of averiguar – verify, to find out
Note: The irregularities are in the uses of the umlaut or dieresis over the u when it is followed by an e; the pronunciation is regular.
Infinitive(infinitivo) averiguar
Gerund (gerundio) averiguando
Participle (participio) averiguado
Present indicative (presente del inidicativo)
yo averiguo, tú averiguas, usted/él/ella averigua
nosotros/as averiguamos, vosotros/as averiguáis, ustedes/ellos/ellas averiguan
Preterite (pretérito) yo averigüé, tu averiguaste, usted/él/ella averiguó
nosotros/as averiguamos, vosotros/as averiguasteis, ustedes/ellos/ellas averiguaron
Imperfect indicativo
(imperfecto del indicativo)
yo averiguaba, tú averiguabas, usted/él/ella averiguaba
nosotros/as averiguábamos, vosotros/as averiguabais,
ustedes/ellos/ellas averiguaban
Future (futuro) yo averiguaré, tú averiguarás, usted/él/ella averiguará
nosotros/as averiguaremos, vosotros/as averiguaréis, ustedes/ellos/ellas averiguarán
13. Conjugation of bendecir – to bless, to consecrate
Note: Also following this pattern is maldecir.
Infinitive (infinitivo) bendecir
Gerund (gerundio) bendiciendo
Participle (participio) bendecido
Present inbendicative(presente del inbendicativo)
yo bendigo, tú bendices, usted/él/ella bendice
nosotros/as bendecimos, vosotros/as bendecís, ustedes/ellos/ellas bendicen
Preterite (pretérito) yo bendije, tu bendijiste, usted/él/ella bendijo nosotros/as bendijimos, vosotros/as bendijisteis, ustedes/ellos/ellas bendijeron
Imperfect inbendicative (imperfecto del inbendicativo)
yo bendecía, tú bendecías, usted/él/ella bendecía
nosotros/as bendecíamos, vosotros/as bendecíais, ustedes/ellos/ellas bendecían
Future (futuro) yo bendeciré, tú bendecirás, usted/él/ella bendecirá
nosotros/as bendeciremos, vosotros/as bendeciréis, ustedes/ellos/ellas bendecirán
14. Conjugation of buscar – to look for
The following conjugation pattern is used by many verbs that end in -car. The irregularity is in spelling only, not in pronunciation.
Note: Other verbs following this pattern include acercar, aparcar, aplicar, arrancar, atacar, colocar, comunicar, criticar, chocar, edificar, equivocar, explicar, indicar, marcar, perjudicar, pescar, practicar, provocar, publicar, sacar, secar, suplicar, tocar, and volcar. The irregular forms are in spelling only; the pronunciation follows the regular pattern.
Infinitive (infinitivo) buscar
Gerund (gerundio) buscando
Participle (participio) buscado
Present indicative (presente del inidicativo)
yo busco, tú buscas, usted/él/ella busca
nosotros/as buscamos, vosotros/as buscáis, ustedes/ellos/ellas buscan
Preterite (pretérito) yo busqué, tu buscaste, usted/él/ella buscó
nosotros/as buscamos, vosotros/as buscasteis, ustedes/ellos/ellas buscaron
Imperfect indicativo
(imperfecto del indicativo) yo buscaba, tú buscabas, usted/él/ella buscaba nosotros/as buscábamos, vosotros/as buscabais, ustedes/ellos/ellas buscaban
Future (futuro) yo buscaré, tú buscarás, usted/él/ella buscará
nosotros/as buscaremos, vosotros/as buscaréis, ustedes/ellos/ellas buscarán
15. Conjugation of avergonzar – to feel ashame, to be ashame
Infinitivo - avergonzar
Gerund (gerundio) - avergonzando
Participle (participio) - avergonzado
Present indicative (presente del inidicativo)
yo avergüenzo, tú avergüenzas, usted/él/ella avergüenza
nosotros/as avergonzamos, vosotros/as avergonzáis, ustedes/ellos/ellas avergüenzan
Preterite (pretérito) yo avergoncé, tu avergonzaste, usted/él/ella avergonzó
nosotros/as avergonzamos, vosotros/as avergonzasteis,
ustedes/ellos/ellas avergonzaron
Imperfect indicativo (imperfecto del indicativo)
yo avergonzaba, tú avergonzabas, usted/él/ella avergonzaba
nosotros/as avergonzábamos, vosotros/as avergonzabais,
ustedes/ellos/ellas avergonzaban
Future (futuro) yo avergonzaré, tú avergonzarás, usted/él/ella avergonzará, nosotros/as avergonzaremos, vosotros/as avergonzaréis,
ustedes/ellos/ellas avergonzarán
16. Conjugation of caber – to fit, to be capable
Infinitive (infinitivo) caber
Gerund (gerundio) cabiendo
Participle (participio) cabido
Present indicative (presente del indicativo)
yo quepo, tú cabes, usted/él/ella cabe
nosotros/as cabemos, vosotros/as cabéis, ustedes/ellos/ellas caben
Preterite (pretérito) yo cupe, tu cupiste, usted/él/ella cupo
nosotros/as cupimos, vosotros/as cupisteis, ustedes/ellos/ellas cupieron
Imperfect indicativo
(imperfecto del indicativo)yo cabía, tú cabías, usted/él/ella cabía
nosotros/as cabíamos, vosotros/as cabíais, ustedes/ellos/ellas cabían
Future (futuro) yo cabré, tú cabrás, usted/él/ella cabrá
nosotros/as cabremos, vosotros/as cabréis, ustedes/ellos/ellas cabrán
17. Caer – to fall
Infinitive(infinitivo) Caer
Gerund (gerundio) cayendo
Participle (participio) caído
Presente yo caigo, tú caes, usted/él/ella cae
nosotros/as caemos, vosotros/as caéis, ustedes/ellos/ellas caen
Preterite (pretérito) yo caí, tu caíste, usted/él/ella cayó
nosotros/as caímos, vosotros/as caísteis, ustedes/ellos/ellas cayeron
Imperfect indicativo (imperfecto del indicativo)
yo caía, tú caías, usted/él/ella caía
nosotros/as caíamos, vosotros/as caíais, ustedes/ellos/ellas caían
Future (futuro) yo caeré, tú caerás, usted/él/ella caerá
nosotros/as caeremos, vosotros/as caeréis, ustedes/ellos/ellas caerán
18. Conjugation of cocer – to cook, to boil, to bake
Note: Torcer also follows this pattern.
Infinitive (infinitivo) cocer
Gerund (gerundio) cociendo
Participle (participio) cocido
Present indicative (presente del indicativo)
yo cuezo, tú cueces, usted/él/ella cuece
nosotros/as cocemos, vosotros/as cocéis, ustedes/ellos/ellas cuecen
Preterite (pretérito) yo cocí, tu cociste, usted/él/ella coció
nosotros/as cocimos, vosotros/as cocisteis, ustedes/ellos/ellas cocieron
Imperfect indicativo (imperfecto del indicativo)
yo cocía, tú cocías, usted/él/ella cocía
nosotros/as cocíamos, vosotros/as cocíais, ustedes/ellos/ellas cocían
Future (futuro) yo coceré, tú cocerás, usted/él/ella cocerá
nosotros/as coceremos, vosotros/as coceréis, ustedes/ellos/ellas cocerán
19. Conjugation of coger – to pick, to take hold, to catch
Note: Other verbs following this pattern include encoger, escoger, proteger, and recoger. The differences from the regular conjugation are in spelling only, not pronunciation.
Infinitive (infinitivo) coger
Gerund (gerundio) cogiendo
Participle (participio) cogido
Present indicative (presente del indicativo)
yo cojo, tú coges, usted/él/ella coge
nosotros/as cogemos, vosotros/as cogéis, ustedes/ellos/ellas cogen
Preterite (pretérito) yo cogí, tu cogiste, usted/él/ella cogió
nosotros/as cogimos, vosotros/as cogisteis, ustedes/ellos/ellas cogieron
Imperfect indicativo
(imperfecto del indicativo) yo cogía, tú cogías, usted/él/ella cogía nosotros/as cogíamos, vosotros/as cogíais, ustedes/ellos/ellas cogían
Future (futuro) yo cogeré, tú cogerás, usted/él/ella cogerá
nosotros/as cogeremos, vosotros/as cogeréis, ustedes/ellos/ellas cogerán
20. Conjugation of comenzar – to begin, to commence
Note: Other verbs following this pattern include empezar and tropezar.
Infinitive (infinitivo) comenzar
Gerund (gerundio) comenzando
Participle (participio) comenzado
Present indicative (presente del inidicativo)
yo comienzo, tú comienzas, usted/él/ella comienza
nosotros/as comenzamos, vosotros/as comenzáis, ustedes/ellos/ellas comienzan
Preterite (pretérito) yo comencé, tu comenzaste, usted/él/ella comenzó nosotros/as comenzamos, vosotros/as comenzasteis, ustedes/ellos/ellas comenzaron
Imperfect indicativo (imperfecto del indicativo)
yo comenzaba, tú comenzabas, usted/él/ella comenzaba
nosotros/as comenzábamos, vosotros/as comenzabais, ustedes/ellos/ellas comenzaban
Future (futuro) yo comenzaré, tú comenzarás, usted/él/ella comenzará
nosotros/as comenzaremos, vosotros/as comenzaréis, ustedes/ellos/ellas comenzarán
21. Conjugation of conducer – to drive, to lead
Note: Other verbs following this pattern include introducir, producir, reducir, reproducir, seducir, and traducir.
Infinitive(infinitivo) conducir
Gerund (gerundio) conduciendo
Participle (participio) conducido
Present indicative (presente del indicativo)
yo conduzco, tú conduces, usted/él/ella conduce
nosotros/as conducimos, vosotros/as conducís, ustedes/ellos/ellas conducen
Preterite (pretérito) yo conduje, tu condujiste, usted/él/ella condujo
nosotros/as condujimos, vosotros/as condujisteis, ustedes/ellos/ellas condujeron
Imperfect indicative (imperfecto del indicativo)
yo conducía, tú conducías, usted/él/ella conducía
nosotros/as conducíamos, vosotros/as conducíais, ustedes/ellos/ellas conducían
Future (futuro) yo conduciré, tú conducirás, usted/él/ella conducirá
nosotros/as conduciremos, vosotros/as conduciréis, ustedes/ellos/ellas conducirán
22. Conjugation of conocer – to know, to meet
This conjugation pattern also is used by many other verbs that end in -cer.
Note: Other verbs following this pattern include agradecer, complacer, crecer, desconocer, desobedecer, florecer, merecer, nacer, obedecer, ofrecer, perecer, pertenecer, and reconocer.
Infinitive (infinitivo) conocer
Gerund (gerundio) conociendo
Participle (participio) conocido
Present indicative (presente del indicativo)
yo conozco, tú conoces, usted/él/ella conoce
nosotros/as conocemos, vosotros/as conocéis, ustedes/ellos/ellas conocen
Preterite (pretérito) yo conocí, tu conociste, usted/él/ella conoció
nosotros/as conocimos, vosotros/as conocisteis, ustedes/ellos/ellas conocieron
Imperfect indicativo
(imperfecto del indicative) yo conocía, tú conocías, usted/él/ella conocía nosotros/as conocíamos, vosotros/as conocíais, ustedes/ellos/ellas conocían
Future (futuro) yo conoceré, tú conocerás, usted/él/ella conocerá
nosotros/as conoceremos, vosotros/as conoceréis, ustedes/ellos/ellas conocerán
23. Conjugation of contar – to tell, to count
Note: Other verbs following tghis pattern include acordar, acostar, apostar, comprobar, consolar, costar, demostrar, encontrar, mostrar, probar, recordar, soltar, sonar, soñar, and volar.
Infinitive(infinitivo) contar
Gerund (gerundio) contando
Participle (participio) contado
Present indicative (presente del inidicativo)
yo cuento, tú cuentas, usted/él/ella cuenta
nosotros/as contamos, vosotros/as contáis, ustedes/ellos/ellas cuentan
Preterite (pretérito) yo conté, tu contaste, usted/él/ella contó
nosotros/as contamos, vosotros/as contasteis, ustedes/ellos/ellas contaron
Imperfect indicativo
(imperfecto del indicative) yo contaba, tú contabas, usted/él/ella contaba nosotros/as contábamos, vosotros/as contabais, ustedes/ellos/ellas contaban
Future (futuro) yo contaré, tú contarás, usted/él/ella contará
nosotros/as contaremos, vosotros/as contaréis, ustedes/ellos/ellas contarán
24. Conjugation of continuar - to continue
Note: Also matching this pattern is actuar.
Infinitive(infinitivo) continuar
Gerund (gerundio) continuando
Participle (participio) continuado
Present indicative (presente del inidicativo)
yo continúo, tú continúas, usted/él/ella continúa
nosotros/as continuamos, vosotros/as continuáis, ustedes/ellos/ellas continúan
Preterite (pretérito) yo continué, tu continuaste, usted/él/ella continuó nosotros/as continuamos, vosotros/as continuasteis, ustedes/ellos/ellas continuaron
Imperfect indicativo (imperfecto del indicativo)
yo continuaba, tú continuabas, usted/él/ella continuaba
nosotros/as continuábamos, vosotros/as continuabais,
ustedes/ellos/ellas continuaban
Future (futuro) yo continuaré, tú continuarás, usted/él/ella continuará
nosotros/as continuaremos, vosotros/as continuaréis, ustedes/ellos/ellas continuarán
25. Conjugation of convertir – to convert
This conjugation pattern is also used for several other verbs.
Note: Other verbs following this pattern include adherir, advertir, arrepentir, convertir, digerir, divertir, herir, hervir, invertir, and referir.
Infinitive(infinitivo) convertir
Gerund (gerundio) convertiendo
Participle (participio) convertido
Present indicative (presente del indicativo)
yo convierto, tú conviertes, usted/él/ella convierte
nosotros/as convertimos, vosotros/as convertís, ustedes/ellos/ellas convierten
Preterite (pretérito) yo convertí, tu convertiste, usted/él/ella convertió
nosotros/as convertimos, vosotros/as convertisteis, ustedes/ellos/ellas convirtieron
Imperfect indicative (imperfecto del indicativo)
yo convertía, tú convertías, usted/él/ella convertía
nosotros/as convertíamos, vosotros/as convertíais, ustedes/ellos/ellas convertían
Future (futuro) yo convertiré, tú convertirás, usted/él/ella convertirá
nosotros/as convertiremos, vosotros/as convertiréis, ustedes/ellos/ellas convertirán
26. Conjugation of creer – to believe
This conjugation pattern is also used by verbs such as creer and poseer.
Note: The irregularity of creer is a matter of spelling, not pronunciation. Other verbs following this pattern are leer and poseer.
Infinitive (infinitivo) creer
Gerund (gerundio) creyendo
Participle (participio) creído
Present indicative (presente del indicativo)
yo creo, tú crees, usted/él/ella cree
nosotros/as creemos, vosotros/as creéis, ustedes/ellos/ellas creen
Preterite (pretérito) yo creí, tu creíste, usted/él/ella creyó
nosotros/as creímos, vosotros/as creísteis, ustedes/ellos/ellas creyeron
Imperfect indicativo
(imperfecto del indicativo) yo creía, tú creías, usted/él/ella creía
nosotros/as creíamos, vosotros/as creíais, ustedes/ellos/ellas creían
Future (futuro) yo creeré, tú creerás, usted/él/ella creerá
nosotros/as creeremos, vosotros/as creeréis, ustedes/ellos/ellas creerán
27. Conjugation of dar – to give
Infinitive(infinitivo) dar
Gerund (gerundio) dando
Participle (participio) dado
Present indicative (presente del inidicativo)
yo doy, tú das, usted/él/ella da
nosotros/as damos, vosotros/as daís, ustedes/ellos/ellas dan
Preterite (pretérito) yo di, tu diste, usted/él/ella dio
nosotros/as dimos, vosotros/as disteis, ustedes/ellos/ellas dieron
Imperfect indicativo
(imperfecto del indicativo) yo daba, tú dabas, usted/él/ella daba
nosotros/as dábamos, vosotros/as dabais, ustedes/ellos/ellas daban
Future (futuro) yo daré, tú darás, usted/él/ella dará
nosotros/as daremos, vosotros/as daréis, ustedes/ellos/ellas darán
28. Conjugation of delinquir - to commit a crime
Note: This verb is regular in its pronunciation but irregular in its spelling.
Infinitive (infinitivo) delinquir
Gerund (gerundio) delinquiendo
Participle (participio) delinquido
Present indicative (presente del indicativo)
yo delinco, tú delinques, usted/él/ella delinque
nosotros/as delinquimos, vosotros/as delinquís, ustedes/ellos/ellas delinquen
Preterite (pretérito) yo delinquí, tu delinquiste, usted/él/ella delinquió nosotros/as delinquimos, vosotros/as delinquisteis, ustedes/ellos/ellas delinquieron
Imperfect indicative (imperfecto del indicativo)
yo delinquía, tú delinquías, usted/él/ella delinquía
nosotros/as delinquíamos, vosotros/as delinquíais, ustedes/ellos/ellas delinquían
Future (futuro) yo delinquiré, tú delinquirás, usted/él/ella delinquirá
nosotros/as delinquiremos, vosotros/as delinquiréis, ustedes/ellos/ellas delinquirán
29. Conjugation of Decir – to say , to tell
Forms of decir that are in boldface are irregular. Verbs that are conjugated following the same pattern are condecir, contradecir, desdecir, intradecir and predecir.
Infinitive (infinitivo) - decir
Gerund (gerundio) - diciendo
Participle (participio) - dicho
Present indicative (presente del indicativo)
yo digo, tú dices, usted/él/ella dice
nosotros/as decimos, vosotros/as decís, ustedes/ellos/ellas dicen
Preterite (pretérito)
yo dije, tu dijiste, usted/él/ella dijo
nosotros/as dijimos, vosotros/as dijisteis, ustedes/ellos/ellas dijeron
Imperfect indicative (imperfecto del indicativo)
yo decía, tú decías, usted/él/ella decía, nosotros/as decíamos, vosotros/as decíais, ustedes/ellos/ellas decían
Future (futuro)
yo diré, tú dirás, usted/él/ella dirá, nosotros/as diremos, vosotros/as diréis, ustedes/ellos/ellas dirán
30. Conjugation of dormir - to sleep
Infinitive (infinitivo) dormir
Gerund (gerundio) durmiendo
Participle (participio) dormido
Present indicative (presente del indicativo)
yo duermo, tú duermes, usted/él/ella duerme
nosotros/as dormimos, vosotros/as dormís, ustedes/ellos/ellas duermen
Preterite (pretérito) yo dormí, tu dormiste, usted/él/ella durmió
nosotros/as dormimos, vosotros/as dormisteis, ustedes/ellos/ellas durmieron
Imperfect indicative
(imperfecto del indicativo) yo dormía, tú dormías, usted/él/ella dormía
nosotros/as dormíamos, vosotros/as dormíais, ustedes/ellos/ellas dormían
Future (futuro) yo dormiré, tú dormirás, usted/él/ella dormirá, nosotros/as dormiremos, vosotros/as dormiréis, ustedes/ellos/ellas dormirán
31. Conjugation of dirigir – to direct
Note: Other verbs following this pattern include exigir and surgir. The verb is regular in pronunciation but not in spelling.
Infinitive (infinitivo) dirigir
Gerund (gerundio) dirigiendo
Participle (participio) dirigido
Present indicative (presente del indicativo)
yo dirijo, tú diriges, usted/él/ella dirige
nosotros/as dirigimos, vosotros/as dirigís, ustedes/ellos/ellas dirigen
Preterite (pretérito) yo dirigí, tu dirigiste, usted/él/ella dirigió, nosotros/as dirigimos, vosotros/as dirigisteis, ustedes/ellos/ellas dirigieron
Imperfect indicative (imperfecto del indicativo)
yo dirigía, tú dirigías, usted/él/ella dirigía
nosotros/as dirigíamos, vosotros/as dirigíais, ustedes/ellos/ellas dirigían
Future (futuro) yo dirigiré, tú dirigirás, usted/él/ella dirigirá, nosotros/as dirigiremos, vosotros/as dirigiréis, ustedes/ellos/ellas dirigirán
32. Conjugation of esparcir – to scatter, to spread
Note: This verb is regular in pronunciation but not in spelling.
Infinitive(infinitivo) esparcir
Gerund (gerundio) esparciendo
Participle (participio) esparcido
Present indicative (presente del indicativo)
yo esparzo, tú esparces, usted/él/ella esparce
nosotros/as esparcimos, vosotros/as esparcís, ustedes/ellos/ellas esparcen
Preterite (pretérito) yo esparcí, tu esparciste, usted/él/ella esparció, nosotros/as esparcimos, vosotros/as esparcisteis, ustedes/ellos/ellas esparcieron
Imperfect indicative
(imperfecto del indicativo) yo esparcía, tú esparcías, usted/él/ella esparcía nosotros/as esparcíamos, vosotros/as esparcíais, ustedes/ellos/ellas esparcían
Future (futuro) yo esparciré, tú esparcirás, usted/él/ella esparcirá
nosotros/as esparciremos, vosotros/as esparciréis, ustedes/ellos/ellas esparcirán
33. Conjugation of distinguir – to distinguish
Note: This verb is regular in pronunciation but not in spelling..
Infinitive(infinitivo) distinguir
Gerund (gerundio) distinguiendo
Participle (participio) distinguido
Present indicative (presente del indicativo)
yo distingo, tú distingues, usted/él/ella distingue
nosotros/as distinguimos, vosotros/as distinguís, ustedes/ellos/ellas distinguen
Preterite (pretérito) yo distinguí, tu distinguiste, usted/él/ella distinguió
nosotros/as distinguimos, vosotros/as distinguisteis,
ustedes/ellos/ellas distinguieron
Imperfect indicative (imperfecto del indicativo)
yo distinguía, tú distinguías, usted/él/ella distinguía
nosotros/as distinguíamos, vosotros/as distinguíais,
ustedes/ellos/ellas distinguían
Future (futuro) yo distinguiré, tú distinguirás, usted/él/ella distinguirá
nosotros/as distinguiremos, vosotros/as distinguiréis, ustedes/ellos/ellas distinguirán
34. Conjugation of enviar - to send
Note: Other verbs following this pattern include desviar, fiar, and guiar.
Infinitive(infinitivo) enviar
Gerund (gerundio) enviando
Participle (participio) enviado
Present indicative (presente del inidicativo)
yo envío, tú envías, usted/él/ella envía
nosotros/as enviamos, vosotros/as enviáis, ustedes/ellos/ellas envían
Preterite (pretérito) yo envié, tu enviaste, usted/él/ella envió
nosotros/as enviamos, vosotros/as enviasteis, ustedes/ellos/ellas enviaron
Imperfect indicativo (imperfecto del indicativo)
yo enviaba, tú enviabas, usted/él/ella enviaba
nosotros/as enviábamos, vosotros/as enviabais, ustedes/ellos/ellas enviaban
Future (futuro) yo enviaré, tú enviarás, usted/él/ella enviará
nosotros/as enviaremos, vosotros/as enviaréis, ustedes/ellos/ellas enviarán
35. Conjugation of elegir - to choose
Note: Other verbs following this pattern include corregir, fingir and regir.
Infinitive(infinitivo) elegir
Gerund (gerundio) eligiendo
Participle (participio) elegido
Present indicative (presente del indicativo) yo elijo, tú eliges, usted/él/ella elige, nosotros/as elegimos, vosotros/as elegís, ustedes/ellos/ellas eligen
Preterite (pretérito) yo elegí, tu elegiste, usted/él/ella elegió, nosotros/as elegimos, vosotros/as elegisteis, ustedes/ellos/ellas eligieron
Imperfect indicative (imperfecto del indicativo)
yo elegía, tú elegías, usted/él/ella elegía
nosotros/as elegíamos, vosotros/as elegíais, ustedes/ellos/ellas elegían
Future (futuro) yo elegiré, tú elegirás, usted/él/ella elegirá
nosotros/as elegiremos, vosotros/as elegiréis, ustedes/ellos/ellas elegirán
36. Conjugation of entender – to understand
Note: Other verbs following this pattern include ascender, atender, defender, descender, and perder.
Infinitive(infinitivo) entender
Gerund (gerundio) entendiendo
Participle (participio) entendido
Present indicative (presente del indicativo)
yo entiendo, tú entiendes, usted/él/ella entiende
nosotros/as entendemos, vosotros/as entendéis, ustedes/ellos/ellas entienden
Preterite (pretérito) yo entendí, tu entendiste, usted/él/ella entendió nosotros/as entendimos, vosotros/as entendisteis, ustedes/ellos/ellas entendieron
Imperfect indicativo (imperfecto del indicativo)
yo entendía, tú entendías, usted/él/ella entendía
nosotros/as entendíamos, vosotros/as entendíais, ustedes/ellos/ellas entendían
Future (futuro) yo entenderé, tú entenderás, usted/él/ella entenderá
nosotros/as entenderemos, vosotros/as entenderéis, ustedes/ellos/ellas entenderán
37. Conjugation of errar -
Infinitive(infinitivo) errar
Gerund (gerundio) errando
Participle (participio) errado
Present indicative (presente del inidicativo)
yo yerro, tú yerras, usted/él/ella yerra
nosotros/as erramos, vosotros/as erráis, ustedes/ellos/ellas yerran
Preterite (pretérito) yo erré, tu erraste, usted/él/ella erró
nosotros/as erramos, vosotros/as errasteis, ustedes/ellos/ellas erraron
Imperfect indicativo (imperfecto del indicativo)
yo erraba, tú errabas, usted/él/ella erraba
nosotros/as errábamos, vosotros/as errabais, ustedes/ellos/ellas erraban
Future (futuro) yo erraré, tú errarás, usted/él/ella errará
nosotros/as erraremos, vosotros/as erraréis, ustedes/ellos/ellas errarán
38. Conjugation of gozar - to enjoy
Note: Other verbs following this pattern include abrazar, adelgazar, amenazar, aplazar, aterrizar, avanzar, bostezar, cazar, cristalizar, cruzar, descalzar, deslizar, destrozar, disfrazar, economizar, embarazar, encabezar, endulzar, garantizar, izar, lanzar, organizar, realizar, rezar, rechazar, trazar and utulilzar. These verbs are regular in pronunciation. However, the z changes to a c when it comes before an e.
Infinitive (infinitivo) gozar
Gerund (gerundio) gozando
Participle (participio) gozado
Present indicative (presente del inidicativo)
yo gozo, tú gozas, usted/él/ella goza
nosotros/as gozamos, vosotros/as gozáis, ustedes/ellos/ellas gozan
Preterite (pretérito) yo gocé, tu gozaste, usted/él/ella gozó
nosotros/as gozamos, vosotros/as gozasteis, ustedes/ellos/ellas gozaron
Imperfect indicativo
(imperfecto del indicativo) yo gozaba, tú gozabas, usted/él/ella gozaba
nosotros/as gozábamos, vosotros/as gozabais, ustedes/ellos/ellas gozaban
Future (futuro) yo gozaré, tú gozarás, usted/él/ella gozará
nosotros/as gozaremos, vosotros/as gozaréis, ustedes/ellos/ellas gozarán
39. Conjugation of hacer – to make, to do
Note: Other verbs following this pattern are deshacer and satisfacer.
Infinitive(infinitivo) Hacer
Gerund (gerundio) Hacienda
Participle (participio) Hecho
Present indicative (presente del indicativo)
yo hago, tú haces, usted/él/ella hace
nosotros/as hacemos, vosotros/as hacéis, ustedes/ellos/ellas hacen
Preterite (pretérito) yo hice, tu hiciste, usted/él/ella hizo
nosotros/as hicimos, vosotros/as hicisteis, ustedes/ellos/ellas hicieron
Imperfect indicativo (imperfecto del indicativo)
yo hacía, tú hacías, usted/él/ella hacía
nosotros/as hacíamos, vosotros/as hacíais, ustedes/ellos/ellas hacían
Future (futuro) yo haré, tú harás, usted/él/ella hará
nosotros/as haremos, vosotros/as hacéis, ustedes/ellos/ellas harán
40. Conjugation of huir – to flee
Note: Other verbs following this pattern include concluir, construir, contribuir, destruir, influir, and other verbs ending in -uir.
Infinitive (infinitivo) Huir
Gerund (gerundio) Huyendo
Participle (participio) Huido
Present indicative (presente del indicativo)
yo huyo, tú huyes, usted/él/ella huye
nosotros/as huimos, vosotros/as huís, ustedes/ellos/ellas huyen
Preterite (pretérito) yo huí, tu huiste, usted/él/ella huyó
nosotros/as huimos, vosotros/as huisteis, ustedes/ellos/ellas huyeron
Imperfect indicative (imperfecto del indicativo)
yo huía, tú huías, usted/él/ella huía
nosotros/as huíamos, vosotros/as huíais, ustedes/ellos/ellas huían
Future (futuro) yo huiré, tú huirás, usted/él/ella huirá
nosotros/as huiremos, vosotros/as huiréis, ustedes/ellos/ellas huirán
41. Conjugation of oir - to hear
Infinitive (infinitivo) oír
Gerund (gerundio) oyendo
Participle (participio) oído
Present indicative
(presente del indicativo) yo oigo, tú oyes, usted/él/ella oye
nosotros/as oímos, vosotros/as oís, ustedes/ellos/ellas oyen
Preterite (pretérito) yo oí, tu oíste, usted/él/ella oyó
nosotros/as oímos, vosotros/as oísteis, ustedes/ellos/ellas oyeron
Imperfect indicative
(imperfecto del indicativo) yo oía, tú oías, usted/él/ella oía
nosotros/as oíamos, vosotros/as oíais, ustedes/ellos/ellas oían
Future (futuro) yo oiré, tú oirás, usted/él/ella oirá
nosotros/as oiremos, vosotros/as oiréis, ustedes/ellos/ellas oirán
42. Conjugation of teñir - to dye
Infinitive (infinitivo) teñir
Gerund (gerundio) tiñendo
Participle (participio) teñido o tinto
Present indicative (presente del indicativo)
yo tiño, tú tiñes, usted/él/ella tiñe
nosotros/as teñimos, vosotros/as teñís, ustedes/ellos/ellas tiñen
Preterite (pretérito) yo teñí, tu teñiste, usted/él/ella tiñó
nosotros/as teñimos, vosotros/as teñisteis, ustedes/ellos/ellas tiñeron
Imperfect indicative
(imperfecto del indicativo) yo teñía, tú teñías, usted/él/ella teñía nosotros/as teñíamos, vosotros/as teñíais, ustedes/ellos/ellas teñían
Future (futuro) yo tiñiré, tú teñirás, usted/él/ella teñirá
nosotros/as teñiremos, vosotros/as teñiréis, ustedes/ellos/ellas teñirán
43. Conjugation of volver – to return
Note: Other verbs following this pattern are absolver, devolver, disolver, desenvolver, resolver, and revolver.
Infinitive (infinitivo) volver
(to turn, return, go back)
Gerund (gerundio) Volviendo
Participle (participio) Vuelto
Present indicative presente del indicativo)
yo vuelvo, tú vuelves, usted/él/ella vuelve
nosotros/as volvemos, vosotros/as volvéis, ustedes/ellos/ellas vuelven
Preterite (pretérito) yo volví, tu volviste, usted/él/ella volvió
nosotros/as volvimos, vosotros/as volvisteis, ustedes/ellos/ellas volvieron
Imperfect indicativo
(imperfecto del indicativo) yo volvía, tú volvías, usted/él/ella volvía nosotros/as volvíamos, vosotros/as volvíais, ustedes/ellos/ellas volvían
Future (futuro) yo volveré, tú volverás, usted/él/ella volverá
nosotros/as volveremos, vosotros/as volveréis, ustedes/ellos/ellas volverán
44. Conjugation of vestir - to dress
Note: Other verbs conjugated in this pattern include competir, despedir, impedir, medir, pedir and repetir and servir.
Infinitive(infinitivo) vestir
Gerund (gerundio) vistiendo
Participle (participio) vestido
Present indicative (presente del indicativo)
yo visto, tú vistes, usted/él/ella viste
nosotros/as vestimos, vosotros/as vestís, ustedes/ellos/ellas visten
Preterite (pretérito) yo vestí, tu vestiste, usted/él/ella vistió
nosotros/as vestimos, vosotros/as vestisteis, ustedes/ellos/ellas vistieron
Imperfect indicative (imperfecto del indicativo)
yo vestía, tú vestías, usted/él/ella vestía
nosotros/as vestíamos, vosotros/as vestíais, ustedes/ellos/ellas vestían
Future (futuro) yo vestiré, tú vestirás, usted/él/ella vestirá
nosotros/as vestiremos, vosotros/as vestiréis, ustedes/ellos/ellas vestirán
45. Conjugation of valer – to be worth
Infinitive(infinitivo) valer
Gerund (gerundio) valiendo
Participle (participio) valido
Present indicative (presente del indicativo)
yo valgo, tú vales, usted/él/ella vale
nosotros/as valemos, vosotros/as valéis, ustedes/ellos/ellas valen
Preterite (pretérito) yo valí, tu valiste, usted/él/ella valió
nosotros/as valimos, vosotros/as valisteis, ustedes/ellos/ellas valieron
Imperfect indicativo (imperfecto del indicativo)
yo valía, tú valías, usted/él/ella valía
nosotros/as valíamos, vosotros/as valíais, ustedes/ellos/ellas valían
Future (futuro) yo valdré, tú valdrás, usted/él/ella valdrá
nosotros/as valdremos, vosotros/as valdréis, ustedes/ellos/ellas valdrán
46. Conjugation of vencer - to conquer
Note: Also following this pattern are mecer and convencer. This pattern is regular in pronunciation but not in spelling.
Infinitive(infinitivo) vencer
Gerund (gerundio) venciendo
Participle (participio) vencido
Present indicative (presente del indicativo)
yo venzo, tú vences, usted/él/ella vence
nosotros/as vencemos, vosotros/as vencéis, ustedes/ellos/ellas vencen
Preterite (pretérito) yo vencí, tu venciste, usted/él/ella venció
nosotros/as vencimos, vosotros/as vencisteis, ustedes/ellos/ellas vencieron
Imperfect indicativo (imperfecto del indicativo)
yo vencía, tú vencías, usted/él/ella vencía
nosotros/as vencíamos, vosotros/as vencíais, ustedes/ellos/ellas vencían
Future (futuro) yo venceré, tú vencerás, usted/él/ella vencerá
nosotros/as venceremos, vosotros/as venceréis, ustedes/ellos/ellas vencerán
47. Conjugation of romper - to break
Infinitive(infinitivo) romper
Gerund (gerundio) rompiendo
Participle (participio) roto
Present indicative (presente del indicativo)
yo rompo, tú rompes, usted/él/ella rompe
nosotros/as rompemos, vosotros/as rompéis, ustedes/ellos/ellas rompen
Preterite (pretérito) yo rompí, tu rompiste, usted/él/ella rompió
nosotros/as rompimos, vosotros/as rompisteis, ustedes/ellos/ellas rompieron
Imperfect indicativo (imperfecto del indicativo)
yo rompía, tú rompías, usted/él/ella rompía
nosotros/as rompíamos, vosotros/as rompíais, ustedes/ellos/ellas rompían
Future (futuro) yo romperé, tú romperás, usted/él/ella romperá
nosotros/as romperemos, vosotros/as romperéis, ustedes/ellos/ellas romperán
This site is use for posting subject related requirements for my classes at the NOTRE DAME OF MIDSAYAP COLLEGE (NDMC) in Midsayap, Cotabato, Philippines. Student can access through this site during their free time and to be updated with what's going on in their respective subject under me.
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Wednesday, September 15, 2010
Thursday, June 17, 2010
Disrete Structure Lesson 1
CSE/IT/IS 214/Math E 211 – DISCRETE STRUCTURES ( Math 217 old)
References:
R1 Kolman, Bernard and Busby, Robert . ( 1987 ) .Discrete Mathematical Structures for
Computer Science, 2nd ed. . London : Prentice Hall International
R2 Johnsonbaugh, Richard . ( 1993 ) . Discrete Mathematics, 3rd ed.. Macmillian Publishing Company
R3 Kolman, Bernard , Busby, Robert C. and Ross, Sharon Cutler . ( 2000 ) . Discrete
Mathematical Structures, 4th ed. . Upper Saddle River, New Jersey : Prentice-Hall Inc.
CHAPTER I. LOGIC AND PROOFS
LOGIC is the discipline that deals with reasoning. On an elementary level, logic provides rules and techniques for determining whether a given argument is valid. Logical reasoning is used in mathematics to prove theorems, in computer science, to verify the correctness of programs and to prove theorems, in natural and physical sciences to draw conclusions from experiments, in social sciences, and in our everyday lives, to solve a multitude of problems
A STATEMENT or a PROPOSITION is a declarative sentence that is either true or false but not both. It is typically expressed as a declarative sentence (as opposed to question or command ). Propositions are the basic building blocks of any theory of logic.
Which of the following are statements or proposition ?
1. The only positive integer that divide 7 are 1 and 7 itself.
2. 2 + 4 = 6
3. The earth is round.
4. 4 – x = 7.
5. Do you speak Chinese ?
6. Take two paracetamol.
7. Buy three tickets for the Manny Pacquiao bout on June 29, 2008.
8. The only positive integer that divide 12 are 3, 4 and itself.
9. A square is a rectangle having all sides equal.
10. Please come on time everyday to avoid being late.
11. Why is it required to enroll prerequisite subjects first ?
12. The rock samples from the moon were studied in the NASA laboratory.
Answers :
1. true, n is prime if n > 1 and can only be divided by 1 and itself ( statement )
2. true. 3. true ( statement )
4. is a declarative sentence, but not a statement, since it is true or false depending on the
value of x that is used.
5. is a question, so it is not a statement.
6. it is not a statement, it is a command.
7. is neither true nor false, it is a command.
LOGICAL CONNECTIVES AND COMPOUND STATEMENTS
The Connectives are AND, OR, NOT, IF...THEN, and IF and ONLY IF
Let us define the meaning of these connectives by showing the relationship between the truth value (i.e. true or false) of composite propositions and those of their component propositions. They are going to be shown using truth table In the tables P and Q represent arbitrary propositions, and true and false are represented by T and F, respectively.
In mathematics, the letters x, y, z . . . denote variables that can be replaced by real numbers, and they can be combined by the operations + , x , – , and . In logic, the letters p, q, r . . . denote propositional variables, that is, variables that can be replaced by statements. Statements or propositional variables can be combined by logical connectives to obtain compound statements or propositions..
Definition 1.1.1 : Let p and q be propositions.
The conjunction of p and q, denoted by p q, is the proposition p and q.
The disjunction of p and q, denoted by p V q, is the proposition p or q.
The negation of p, denoted by p or ( P), is the proposition not p.
Propositions such as p q and p V q that result from combining propositions are called compound propositions. The compound statement p q is true when both p and q are true; otherwise, it is false. The compound statement p V q is true if at least one of p or q is true, it is false when both p and q are false.
The truth values of propositions such as conjunctions and disjunctions can be described by truth tables. The truth table of a proposition p made up of the individual propositions p1. . . pn, lists all possible combinations of truth values for p1. . . pn, T denoting true and F denoting false and for each such combination lists the truth value of p.
Truth Tables
P AND Q ( P Q ) P OR Q (P V Q ) not p ( p or P )
P Q P Q P Q P V Q p ( P)
T T T T T T T F
T F F T F T F T
F T F F T T
F F F F F F
AND. The table shows that (P Q) is true if both P and Q are true, and that it is false in any other case.
NOT: The table shows that if P is true, then (~P) is false, and that if P is false, then (~P) is true.
Examples :
1. p : 2 + 3 = 7 , q : A century is 100 years.
a) P is false and q is true. The conjunction p q : 2 + 3 = 7 and a century is 100 years is false.
b) The disjunction p V q : 2 + 3 = 7 and a century is 100 years is true.
2. p : Kidapawan City is the capital of Cotabato. , q : Cotabato is in Mindanao.
a) P is true and q is true. The conjunction p q : Kidapawan City is the capital of
Cotabato and Cotabato is in Mindanao is true.
b) The disjunction p V q : The conjunction p q : Kidapawan City is the capital of
Cotabato and Cotabato is in Mindanao is true.
3. p : i 2 = –1 , q : e = 2.7182818
4. P : i = √–1 , q : The seventh month is August.
5. P : The 2nd day of the week is Friday , q : June 12 is labor day.
6. p : Rizal is from Leyte. , q : Dagohoy is from Bohol
7. P : Snakes are reptiles. , q : Birds are mammals.
Example : P : Discrete mathematics is easy. Not P : Discrete mathematics is not easy.
P : The price of rice is increasing. Not P : The price of rice is not increasing.
Evaluate the truth value for each proposition if the statements p = F , q = T and r = F.
1. p V q 4. p V r 7. p r 10. p q
2. ~P V r 5. r V q 8. r q 11. ~P r
3. ~P V ~q 6. p V ~r 9. p ~r 12. ~P ~q
Write the truth table for each proposition.
1. p ~q 2. ( p V q ) ~ p 3. (~p V ~q ) V p 4. ( p q ) V (~p V q )
Determine whether each statement is True or False.
1. 3 < 7 and 7 < 9 2. It is not the case that ( 3 < 7 and 7 < 9 )
Formulate the symbolic expression in words using p : Today is Friday,
q : It is sunny, and r : It is cold.
1. p V q 2. ~ ( p V q ) r 3. ( p q ) (~ ( r V p ) ) 4. ( p r ) V q
QUANTIFIERS
We used the notation P ( x ) to denote a sentence or a statement P concerning the variable object x. The set defined by P ( x ), written { x | P ( x ) }, is a collection of all objects for which P is sensible and true. For example, { x | x is a positive integer less than 4} is the set { 1, 2, 3 } described by listing its elements.
A sentence P ( x ) is also called predicate, because in English the property is grammatically a predicate. P ( x ) is also called a propositional function, because each choice of x produces a proposition P ( x ) that is either true or false. Another use of predicate is in programming. Two common constructions are “ if P ( x ), then execute certain steps”, and “ while Q( x ), do specified actions”. The predicates P ( x ) and Q ( x ) are called the guards for the block of programming code. Often the guard for block is a conjunction or disjunction.
Example : Let A = { x | x is an integer less than 6 }. Here P ( x ) is the sentence “ x is an integer less than 6.” The common property is “ is an integer less than 6”. Since P ( 3 ) is true, 3 A , P ( 7 ) is false, 7 A.
1. Universal Quantifier
The Universal Quantifier of a predicate P ( x ) is the statement “For all values of x, P ( x ) is true”. It is assumed that only values of x that make sense in P ( x ) are considered. The universal quantification P ( x ) is denoted by x P ( x ). The symbol is called a universal quantifier. Universal quantification can also be stated as “For every x”, “Every x”, or “For any x”.
Example 2. The sentence P ( x ) : – ( – x ) = x is a predicate that make sense for real numbers x.
The universal quantification of P ( x ), x P(x), is a true statement, because for all real numbers, – ( – x ) = x.
Example 3. Let Q ( x ) : x + 3 7. Then x Q ( x ), is a false statement because Q ( 6 ), is not true.
A predicate may contain several variables. Universal quantification maybe applied to each of the variables. For example the commutative property can be expressed as x y x y = y x. The order in which the universal quantifier are considered does not change the truth value. Often mathematical statements contain implied universal quantification.
Example 4. Commutative property on sets:
1. A B = B A 2. A B = B A
2. Existential Quantifier
The Existential Quantification of a predicate P ( x ) is the statement “There exist a value of x for which P ( x ) is true”. The existential quantification of P ( x ) is denoted by x P ( x ). The symbol is called the existential quantifier. x can also be read as “There exist an x”, “There is some x”, “There exist an x”, or “There is at least one x”.
Example 5. Let Q ( x ) : x + 2 5. The existential quantification of Q ( x ), x Q( x ) is a true
statement because Q (1) is a true statement. Q (4) is false.
Example 6. The statement y y + 4 = y is false. There is no value of y for which the proportional
function y + 4 = y produces a true statement.
More examples.
1. Let p: For all positive integers n, n2 + 41 n + 41 is a prime number.
Then ~p is : There is at least one positive integer n for which n2 + 41n + 41 is not prime.
2. Let q : There is some integer k for which 12 = 3k. Then ~q : For all integers k, 12 3k.
Exercises from reference R3.
P ( x ) : x is even, Q( x ): x is a prime number, R ( x , y ) : x + y is even. The variables x and y are
integers. Write an English sentence corresponding to the following :
[14/51] a) x P(x) b) x Q( x )
[15/51] a) x y R( x , y ) b) x y R( x , y )
[16/51] a) x (~Q( x ) ) b) y (~ P ( y ) )
[17/51] a) ~ ( x ( P ( x ) ) b) ~ (x Q( x ) )
CONDITIONAL PROPOSITIONS AND LOGICAL EQUIVALENCE
Definition 1.2.1 : Let p and q be propositions.
If p and q are propositions, the compound proposition if p then q is called a conditional
proposition or implication and is denoted by p q.
The proposition p is called hypothesis ( or antecedent ) and the proposition q is called the conclusion ( or consequent ). The truth value is defined by the following table :
p q p q
T T T
T F F
F T T
F F T
In logic, implication is used in a much weaker sense. To say that the compound statement p q is true simply asserts that if p is true, then q will also be found to be true. In other words, p q says that we will not have p true and q false at the same time. The truth values in the table shows p q in terms of the truth values of p and q. Observed that p q is considered false only if p is true and q is false. In particular, if p is false, then p q is true for any q.
Example :
1. p : I am thirsty 2. p : It is smoking.
q : I will drink. q : 4 + 5 = 9
Answers: 1. If I am thirsty, then I will drink.
2. If it is smoking, then 4 + 5 = 9.
In the English language, and in mathematics, each of the following expressions is an equivalent form of the conditional statement p q :
p implies q
q, if p
p only if q
p is a sufficient condition for q
q is a necessary condition for q
If p q is an implication, then the converse of p q is the implication q p, and the contrapositive of p q is the implication q p.
Examples :
1. Give the converse and contrapositive of the implication “If it is raining, then I get wet”.
Solution : p : It is raining
q : I get wet.
Converse : If I get wet, then it is raining.
Contrapositive : If I do not get wet, then it is not raining.
Definition 1.2.2 : Let p and q be propositions.
If p and q are propositions, the compound proposition p if and only if q denoted by p q is called an equivalence or biconditional proposition. The truth value is defined by the truth table below. Observe that p q is true only when both p and q are True or when both are False. p q is also stated as p is a necessary and sufficient condition for q. Truth table is :
P q p q
T T T
T F F
F T F
F F T
Example : 1. Is the statement 5 > 3 if and only if 0 < 5 – 3 true ?
Let p : 5 > 3 , q : 0 < 5 – 3. Since p is true and q is true, then p q is true.
In general, a compound statement may have many component parts, each of which is itself a statement, represented by some proportional variable. The statement
s : p ( q ( p r ) )
involves three propositions, p, q, and r, each of which may independently be true or false. There are altogether 23 = 8 possible combinations of truth values for p, q, and r, and the truth value for s contains n components statements, there will be 2 n entries needed in the truth table for s.
Step 1. The first n columns of the table are labeled by the component propositional variables.
Further column are constructed for all intermediate combinations of statements,
culminating in the given statement.
Step 2. Under each of the first n headings, we list the 2 n possible n-tuples of truth values of
the component statement s. Each n-tuple is listed on a separate row.
Step 3. For each row we compute, in sequence, all remaining truth values.
Ex. 2 : Compute the truth table of the statement ( p q ) (~ q ~p ). Also ( p q) ( p V q).
Solution : Using steps 1,2 and 3, construct the following table.
p q p q q p q p ( p q) ( q p) p V q ( p q) ( p V q)
T T T F F T T T T
T F F T F F T T F
F T T F T T T T T
F F T T T T T F F
A statement that is true for all possible values of its propositional variables is called a tautology (as in the truth table above).
A statement that is always false is called a contradiction or an absurdity.
A statement that can be either true or false, depending on the truth values of its propositional variables, is called a contingency.
In the example above :
a) The statement ( p q ) ( q p ) is a tautology.
b) The statement p p is an absurdity.
c) The statement ( p q ) ( p V q ) is a contingency.
We have defined a new mathematical structure with two binary operations and one unary operation, [ proposition, , V, ]. It makes no sense to say two propositions are equal; instead we say p and q are logically equivalent, if p q is a tautology. When an equivalence is shown to be a tautology, it means that its two component parts are always either both true or both false, for any values of the proportional variables. Thus the two sides are simply different ways of making the same statement and can be regarded as “equal”. We denote that p is equivalent to q by p q, now we can adopt our properties for operations to say this structure has a property if using equivalent in place of equal gives a true statement, thus we say that they are logically equivalent.
The binary operation V has the commutative property that is p V q q V p. The truth value for
( p V q ) ( q V p ) shows the statement is a tautology.
p q p V q q V p ( p V q ) ( q V p )
T T T T T
T F T T T
F T T T T
F F F F T
OPERATIONS FOR PROPOSITIONS
Theorem 1. The operations for propositions have the following properties.
Commutative properties Distributive properties
1. p V q q V p 5. p V ( q r ) ( p V q ) ( p V r )
2. p q q p 6. p ( q V r ) ( p q ) V ( p r )
Associative properties Idempotent properties
3. p V ( q V r ) ( p V q ) V r 7. p V p p
4. p ( q r ) ( p q ) r 8. p p p
Properties of Negation
9. ~ ( ~ p ) p 10. ~ ( p V q ) ( ~ p ) ( ~ q ) 11. ~ ( p q ) ( ~ p ) V ( ~ q )
Theorem 2 :
a) ( p q ) ((~ p ) V q ) d) ~ ( p q ) ( p ~ q )
b) ( p q ) (~ q ~ p ) e) ~ ( p q ) [ ( p ~ q ) V (q ~ p ) ]
c) ( p q ) [ ( p q ) (q p ) ]
Proof of Theorem 2a using truth table Proof of Theorem 2b using truth table
p q ~p p q ~ p V q ~q ~p p q ~ q ~ p
T T F T T F F T T
T F F F F T F F F
F T T T T F T T T
F F T T T T T T T
Theorem 3 :
a) ~ ( x P( x ) ) x ~ P( x )
b) ~ ( x P( x )) x ~ ( P( x ) )
c) x P( x ) Q( x ) x P( x ) x Q ( x )
d) x P( x ) x Q ( x ) x P( x ) Q ( x )
e) x ( P( x ) V Q ( x ) ) x P( x ) V x Q ( x )
f) x P( x ) Q ( x ) x P( x ) x Q ( x )
g) ( ( x P( x ) ) V ( x Q ( x )) ) x P( x ) V Q ( x ) ( is a tautology )
h) x ( P( x ) Q ( x ) ) x P( x ) x Q ( x ) ( is a tautology )
Theorem 4 : Each of the following is a tautology.
a) p q p f) ~ ( p q ) p
b) p q q g) ( p ( p q ) ) q
c) p p V q h) ( ~ p ( p V q ) q
d) q ( p V q ) i) ( ~ q ( p q ) ) ~ q
e) ~ p ( p q ) j) ( ( p q ) ) ( q r )) ( p r )
Methods of proof
If an implication p q is a tautology, where p and q maybe compound statements involving any number of propositional variables, we say that q logically follows from p. Suppose an implication of the form ( p1 p2 p3 … pn ) q is a tautology, this implication is true regardless of the truth values of any of its components. Then we can say that q logically follows from p1, p2, . . . , pn. When q logically follows from p1, p2, . . . , pn then we write
p1
p2
.
.
.
pn .
q
Virtually all mathematical theorems are composed of implications of the type ( p1 p2 p3 … pn ) q. The pi’s are called the hypotheses, or premise and q is called the conclusion. To prove the theorem means to show that the implication is a tautology.
Note that we are not trying to show that q is true, but only that q will be true if all the Pi’s are true. For this reason , mathematical proofs often begin with the statement “ Supposed that p1, p2, . . . and pn are true ” and conclude with the statement “ Therefore, q is true.” The proof does not show that q is true but simply show that q has to be true if the Pi’s are all true.
Arguments based on tautologies represent correct methods of reasoning. Their validity depends only on the form of the statements involved and not on the truth values of the variables that they contain. Such arguments are called rules of inference. The various steps in the mathematical proof of a theorem must follow from the use of various rules of inference, and a mathematical proof of a theorem must begin with the hypotheses, proceed through various steps, each justified by some rule of inference, and arrive at the conclusion.
Ex. 1: According to Theorem 4 j [ ( p q ) ( q r ) ] ( p r ) is a tautology.
Thus the argument
p q
q r .
p r is universally valid, and so is a rule of inference.
Ex 2: Is the following argument valid ?
If you invest in the stock market, then you will get rich.
If you get rich, then you will be happy. .
If you invest in the stock market, then you will be happy.
Solution : The argument is of the form given in Ex 1, hence the argument is valid, although
the conclusion may be false.
Ex. 3 : The tautology ( p q ) [ ( p q ) (q p ) ] is Theorem 2c, thus both of the following
arguments are valid :
p q
p q q p
( p q ) (q p ) p q
Some mathematical theorems are equivalences, that is they are of the form p q. They usually stated p if and only if q. By Ex 3, the proof of such theorem is logically equivalent with proving both p q and q p, and this almost always the way in which equivalences are proved. We first assume that p is true, and show that q must then be true; next we assume that q is true and show that p must then be true.
A very important rule of inference is :
p
p q
q That is, p is true, and p q is true, so q is true.
Some rules of inference were given Latin names by classical scholars. Theorem 4g is referred to as modus ponens ( the method of asserting ). Theorem 4g is ( p ( p q ) ) q.
Ex 4: Is the following argument valid ?
Smoking is healthy.
If smoking is healthy, then cigarettes are prescribed by physicians .
Cigarettes are prescribed by physicians.
The argument is valid since it is of the form modus ponens. However, the conclusion is false. Observed that the first premise p: smoking is healthy is false. The second premise p q is then true and ( p ( p q ) ), the conjunction of the two premises, is false.
Ex. 5 : If taxes are lowered, then income rises.
Income rises .
Taxes are lowered.
Solution : Let p : taxes are lowered and q : income rises
The argument is of the form p q
q .
p
Assume that p q and q are both true. Now p q may be true with p being false. Then the conclusion p is false. Hence the argument is not valid.
An important proof technique, called an indirect method of proof , follows from the tautology
( p q ) ( ( ~ q ) ( ~ p ) ). This state, as previously mentioned, that an implication is equivalent to its contrapositive. Thus to prove p q indirectly, we assume q is false ( the statement ~ q ) and show that p is then false ( the statement ~ p ).
Ex. 6. Let n be an integer. Prove that if n2 is odd, then n is odd.
Solution : Let p : n2 is odd and q : n is odd. We have to prove that p q is true. Instead, we prove the contrapositive ~ q ~ p. Suppose that n is not odd, so that n is even. Then n = 2k, where k is an integer . Then, n2 = ( 2k )2 = 4k2 = 2 ( 2k2 ), so n2 is even. Thus, we have shown that if n is even, n2 is even, which is the contrapositive of the given statement. Hence, the given statement has been proved.
Another important proof technique is proof by contradiction. This method is based on the tautology ( ( p q ) ( ~ q ) ) ( ~ p ). Thus the rule of inference
p q
~ q .
~ p is valid. Informally, this states that if a
statement p implies a false statement q, then p must be false. This is often applied to the case where q is an absurdity or contradiction, that is, a statement is always false. An example is given by taking q as the contradiction r ( ~ r ). Thus any statement that implies a contradiction must be false. In order to use the proof by contradiction, suppose we wish to show that a statement q logically follows from statements p1, p2, . . ., pn. Assume that ~q is true ( that is, q is false ) as an extra hypothesis p1 p2 . . . pn ( ~ q ) implies a contradiction, then at least one of the statements p1, p2, . . ., pn, ~ q must be false. This means that if all the pi’s are true, then ~ q must be true. Thus q follows from p1, p2, . . ., pn. This is proof by contradiction.
Ex. 7 : Prove there is no rational number p/q whose square is 2. In other words, show that the
2 is rational.
Solution: This statement is good candidate for proof by contradiction., because we can check all possible rational numbers to demonstrate that none had a square equal to 2. Assume ( p/q )2 = 2 for some integers p and q, which have no common factors. Then p2 = 2q2 , so p2 is even. This implies that p is even, since the square of an odd number is odd. Thus, q2 is even, and so q is even. We now have that both p and q are even, and therefore have a common factor 2. This is a contradiction to the assumption. Thus the assumption must be false.
STEPS IN THE PROOF . . . . .
To prove a theorem of the form ( p1 p2 . . . pn ) q, begin with the hypothesis p1, p2, . . . pn and show that some result r1 logically follows. Then using p1, p2, . . . pn, r1 , show that some other statements r2 logically follows. Continue this process, producing intermediate statements r1, r2, . . ., rk called steps in the proof, until we can finally that the conclusion q logically follows from p1, p2, . . . pn, r1, r2, . . . , rk. Each logical step must be justified by some valid proof technique.
Ex. 8 : Prove or disprove the statement that if x and y are real numbers , ( x2 = y2 ) ( x = y ). Solution : The statement can be restated in the form x y R ( x, y ). To prove the statement, we need to provide steps, each of which would be true for all x and y. To disprove the statement, we
need to find one example for which the implication is false.
Since ( – 3 )2 = 32, but – 3 3, the result is false. This example is called a counterexample, and any other counterexample
would do just as well.
Hence, if a statement claims that a property holds for all objects, then to prove it, we must use steps that are valid for all objects of that type. To disprove such statement, we need only to show one counterexample, that is, one particular object for which the claim fails.
State whether the argument given is valid or not. If it is valid, identify the tautology on which it is based.
( 1/62 ) If I drive to work, then I will arrive tired.
I am not tired when I arrive at work . . Valid. (( drive ) tired ) ~ tired ) ( ~ drive )
I do not drive to work.
( 3/62 ) If I drive to work, then I will arrive tired.
I do not drive to work. . Invalid. ( ( d ) t ) ( ~ d ) ( ~ t ) ( ? )
I will not arrive tired.
( 5/62 ) I will become famous or I will not become a writer.
I will become a writer. . Valid ( ( f V ~ w ) w ) f
I will become famous.
( 7/63 ) If I try hard and I have talent, then I will become a musician.
If I become a musician, then I will be happy. .
If I will not be happy, then I did not try hard or I do not have talent.
Valid : [ ( ht m ) ( m hp )] [ ~ hp ~ ht ]
( 13/63 ) Prove that the sum of two odd numbers is even.
Solution : Suppose m and n are odd numbers. Then there exist integers j and k such that
m = 2j + 1 and n = 2k + 1. m + n = 2j + 1 + 2k + 1 = 2j + 2k + 2 = 2 ( j + k + 1 ).
Since j + k + 1 is an integer, m + n is even.
( 15/63 ) Prove that the structure [ even integers, + , * ] is closed with respect to *.
Solution : Suppose m and n are odd numbers. Then there exist integers j and k such that
m = 2j + 1 and n = 2k + 1. m . n = 2j . 2k + 2j + 2k + 1 = 2 ( 2 jk + j + k ) + 1.
Since 2 jk + j + k is an integer, m . n is odd and the system is closed with respect to multiplication.
( 17/63 ) Prove that A = B if and only if A B and B A.
Solution : If A = B, then, clearly, A B and B A. If A B and B A, then A B A
and B must be the same as A.
( 19/63 ) Show that a) A B is necessary and sufficient condition for A B = B.
b) A B is necessary and sufficient condition for A B = A.
Solution : a) If A B, then A B B. But B A B. Hence A B = B. If A B = B, then
since A A B, we have A B.
b) If A B, then A A B. But A B A. Hence A B = A. If A B = A, then
since A B B, we have A B.
( 21/63 ) Prove or disprove : the sum of any 5 consecutive integers is divisible by 5.
Solution : Any 5 consecutive integers can be represented by n, n + 1, n + 2, n + 3, n + 4.
Their sum is 5 n + 10 or 5 ( n + 2 ). This is clearly divisible by 5.
( 23/63 ) Determine if the following is a valid argument. Explain your conclusion.
Prove : x x3 > x2
Proof : x x2 > 0 so x x2 ( x – 1 ) > 0 ( x – 1 ) and x x3 – x2 > 0. Hence x x3 > x2.
Solution : Invalid. Multiplying by x – 1 may or may not preserve the order of the inequality.
( 27/63 ) prove that the sum of two prime numbers, each larger than 2, is not a prime number.
Solution : Let x and y be the prime numbers, each larger than 2. Then x and y are odd and
their sum is even ( Exer. 13 ). The only even prime is 2, so x + y is not prime.
More . . .
and more !
References:
R1 Kolman, Bernard and Busby, Robert . ( 1987 ) .Discrete Mathematical Structures for
Computer Science, 2nd ed. . London : Prentice Hall International
R2 Johnsonbaugh, Richard . ( 1993 ) . Discrete Mathematics, 3rd ed.. Macmillian Publishing Company
R3 Kolman, Bernard , Busby, Robert C. and Ross, Sharon Cutler . ( 2000 ) . Discrete
Mathematical Structures, 4th ed. . Upper Saddle River, New Jersey : Prentice-Hall Inc.
CHAPTER I. LOGIC AND PROOFS
LOGIC is the discipline that deals with reasoning. On an elementary level, logic provides rules and techniques for determining whether a given argument is valid. Logical reasoning is used in mathematics to prove theorems, in computer science, to verify the correctness of programs and to prove theorems, in natural and physical sciences to draw conclusions from experiments, in social sciences, and in our everyday lives, to solve a multitude of problems
A STATEMENT or a PROPOSITION is a declarative sentence that is either true or false but not both. It is typically expressed as a declarative sentence (as opposed to question or command ). Propositions are the basic building blocks of any theory of logic.
Which of the following are statements or proposition ?
1. The only positive integer that divide 7 are 1 and 7 itself.
2. 2 + 4 = 6
3. The earth is round.
4. 4 – x = 7.
5. Do you speak Chinese ?
6. Take two paracetamol.
7. Buy three tickets for the Manny Pacquiao bout on June 29, 2008.
8. The only positive integer that divide 12 are 3, 4 and itself.
9. A square is a rectangle having all sides equal.
10. Please come on time everyday to avoid being late.
11. Why is it required to enroll prerequisite subjects first ?
12. The rock samples from the moon were studied in the NASA laboratory.
Answers :
1. true, n is prime if n > 1 and can only be divided by 1 and itself ( statement )
2. true. 3. true ( statement )
4. is a declarative sentence, but not a statement, since it is true or false depending on the
value of x that is used.
5. is a question, so it is not a statement.
6. it is not a statement, it is a command.
7. is neither true nor false, it is a command.
LOGICAL CONNECTIVES AND COMPOUND STATEMENTS
The Connectives are AND, OR, NOT, IF...THEN, and IF and ONLY IF
Let us define the meaning of these connectives by showing the relationship between the truth value (i.e. true or false) of composite propositions and those of their component propositions. They are going to be shown using truth table In the tables P and Q represent arbitrary propositions, and true and false are represented by T and F, respectively.
In mathematics, the letters x, y, z . . . denote variables that can be replaced by real numbers, and they can be combined by the operations + , x , – , and . In logic, the letters p, q, r . . . denote propositional variables, that is, variables that can be replaced by statements. Statements or propositional variables can be combined by logical connectives to obtain compound statements or propositions..
Definition 1.1.1 : Let p and q be propositions.
The conjunction of p and q, denoted by p q, is the proposition p and q.
The disjunction of p and q, denoted by p V q, is the proposition p or q.
The negation of p, denoted by p or ( P), is the proposition not p.
Propositions such as p q and p V q that result from combining propositions are called compound propositions. The compound statement p q is true when both p and q are true; otherwise, it is false. The compound statement p V q is true if at least one of p or q is true, it is false when both p and q are false.
The truth values of propositions such as conjunctions and disjunctions can be described by truth tables. The truth table of a proposition p made up of the individual propositions p1. . . pn, lists all possible combinations of truth values for p1. . . pn, T denoting true and F denoting false and for each such combination lists the truth value of p.
Truth Tables
P AND Q ( P Q ) P OR Q (P V Q ) not p ( p or P )
P Q P Q P Q P V Q p ( P)
T T T T T T T F
T F F T F T F T
F T F F T T
F F F F F F
AND. The table shows that (P Q) is true if both P and Q are true, and that it is false in any other case.
NOT: The table shows that if P is true, then (~P) is false, and that if P is false, then (~P) is true.
Examples :
1. p : 2 + 3 = 7 , q : A century is 100 years.
a) P is false and q is true. The conjunction p q : 2 + 3 = 7 and a century is 100 years is false.
b) The disjunction p V q : 2 + 3 = 7 and a century is 100 years is true.
2. p : Kidapawan City is the capital of Cotabato. , q : Cotabato is in Mindanao.
a) P is true and q is true. The conjunction p q : Kidapawan City is the capital of
Cotabato and Cotabato is in Mindanao is true.
b) The disjunction p V q : The conjunction p q : Kidapawan City is the capital of
Cotabato and Cotabato is in Mindanao is true.
3. p : i 2 = –1 , q : e = 2.7182818
4. P : i = √–1 , q : The seventh month is August.
5. P : The 2nd day of the week is Friday , q : June 12 is labor day.
6. p : Rizal is from Leyte. , q : Dagohoy is from Bohol
7. P : Snakes are reptiles. , q : Birds are mammals.
Example : P : Discrete mathematics is easy. Not P : Discrete mathematics is not easy.
P : The price of rice is increasing. Not P : The price of rice is not increasing.
Evaluate the truth value for each proposition if the statements p = F , q = T and r = F.
1. p V q 4. p V r 7. p r 10. p q
2. ~P V r 5. r V q 8. r q 11. ~P r
3. ~P V ~q 6. p V ~r 9. p ~r 12. ~P ~q
Write the truth table for each proposition.
1. p ~q 2. ( p V q ) ~ p 3. (~p V ~q ) V p 4. ( p q ) V (~p V q )
Determine whether each statement is True or False.
1. 3 < 7 and 7 < 9 2. It is not the case that ( 3 < 7 and 7 < 9 )
Formulate the symbolic expression in words using p : Today is Friday,
q : It is sunny, and r : It is cold.
1. p V q 2. ~ ( p V q ) r 3. ( p q ) (~ ( r V p ) ) 4. ( p r ) V q
QUANTIFIERS
We used the notation P ( x ) to denote a sentence or a statement P concerning the variable object x. The set defined by P ( x ), written { x | P ( x ) }, is a collection of all objects for which P is sensible and true. For example, { x | x is a positive integer less than 4} is the set { 1, 2, 3 } described by listing its elements.
A sentence P ( x ) is also called predicate, because in English the property is grammatically a predicate. P ( x ) is also called a propositional function, because each choice of x produces a proposition P ( x ) that is either true or false. Another use of predicate is in programming. Two common constructions are “ if P ( x ), then execute certain steps”, and “ while Q( x ), do specified actions”. The predicates P ( x ) and Q ( x ) are called the guards for the block of programming code. Often the guard for block is a conjunction or disjunction.
Example : Let A = { x | x is an integer less than 6 }. Here P ( x ) is the sentence “ x is an integer less than 6.” The common property is “ is an integer less than 6”. Since P ( 3 ) is true, 3 A , P ( 7 ) is false, 7 A.
1. Universal Quantifier
The Universal Quantifier of a predicate P ( x ) is the statement “For all values of x, P ( x ) is true”. It is assumed that only values of x that make sense in P ( x ) are considered. The universal quantification P ( x ) is denoted by x P ( x ). The symbol is called a universal quantifier. Universal quantification can also be stated as “For every x”, “Every x”, or “For any x”.
Example 2. The sentence P ( x ) : – ( – x ) = x is a predicate that make sense for real numbers x.
The universal quantification of P ( x ), x P(x), is a true statement, because for all real numbers, – ( – x ) = x.
Example 3. Let Q ( x ) : x + 3 7. Then x Q ( x ), is a false statement because Q ( 6 ), is not true.
A predicate may contain several variables. Universal quantification maybe applied to each of the variables. For example the commutative property can be expressed as x y x y = y x. The order in which the universal quantifier are considered does not change the truth value. Often mathematical statements contain implied universal quantification.
Example 4. Commutative property on sets:
1. A B = B A 2. A B = B A
2. Existential Quantifier
The Existential Quantification of a predicate P ( x ) is the statement “There exist a value of x for which P ( x ) is true”. The existential quantification of P ( x ) is denoted by x P ( x ). The symbol is called the existential quantifier. x can also be read as “There exist an x”, “There is some x”, “There exist an x”, or “There is at least one x”.
Example 5. Let Q ( x ) : x + 2 5. The existential quantification of Q ( x ), x Q( x ) is a true
statement because Q (1) is a true statement. Q (4) is false.
Example 6. The statement y y + 4 = y is false. There is no value of y for which the proportional
function y + 4 = y produces a true statement.
More examples.
1. Let p: For all positive integers n, n2 + 41 n + 41 is a prime number.
Then ~p is : There is at least one positive integer n for which n2 + 41n + 41 is not prime.
2. Let q : There is some integer k for which 12 = 3k. Then ~q : For all integers k, 12 3k.
Exercises from reference R3.
P ( x ) : x is even, Q( x ): x is a prime number, R ( x , y ) : x + y is even. The variables x and y are
integers. Write an English sentence corresponding to the following :
[14/51] a) x P(x) b) x Q( x )
[15/51] a) x y R( x , y ) b) x y R( x , y )
[16/51] a) x (~Q( x ) ) b) y (~ P ( y ) )
[17/51] a) ~ ( x ( P ( x ) ) b) ~ (x Q( x ) )
CONDITIONAL PROPOSITIONS AND LOGICAL EQUIVALENCE
Definition 1.2.1 : Let p and q be propositions.
If p and q are propositions, the compound proposition if p then q is called a conditional
proposition or implication and is denoted by p q.
The proposition p is called hypothesis ( or antecedent ) and the proposition q is called the conclusion ( or consequent ). The truth value is defined by the following table :
p q p q
T T T
T F F
F T T
F F T
In logic, implication is used in a much weaker sense. To say that the compound statement p q is true simply asserts that if p is true, then q will also be found to be true. In other words, p q says that we will not have p true and q false at the same time. The truth values in the table shows p q in terms of the truth values of p and q. Observed that p q is considered false only if p is true and q is false. In particular, if p is false, then p q is true for any q.
Example :
1. p : I am thirsty 2. p : It is smoking.
q : I will drink. q : 4 + 5 = 9
Answers: 1. If I am thirsty, then I will drink.
2. If it is smoking, then 4 + 5 = 9.
In the English language, and in mathematics, each of the following expressions is an equivalent form of the conditional statement p q :
p implies q
q, if p
p only if q
p is a sufficient condition for q
q is a necessary condition for q
If p q is an implication, then the converse of p q is the implication q p, and the contrapositive of p q is the implication q p.
Examples :
1. Give the converse and contrapositive of the implication “If it is raining, then I get wet”.
Solution : p : It is raining
q : I get wet.
Converse : If I get wet, then it is raining.
Contrapositive : If I do not get wet, then it is not raining.
Definition 1.2.2 : Let p and q be propositions.
If p and q are propositions, the compound proposition p if and only if q denoted by p q is called an equivalence or biconditional proposition. The truth value is defined by the truth table below. Observe that p q is true only when both p and q are True or when both are False. p q is also stated as p is a necessary and sufficient condition for q. Truth table is :
P q p q
T T T
T F F
F T F
F F T
Example : 1. Is the statement 5 > 3 if and only if 0 < 5 – 3 true ?
Let p : 5 > 3 , q : 0 < 5 – 3. Since p is true and q is true, then p q is true.
In general, a compound statement may have many component parts, each of which is itself a statement, represented by some proportional variable. The statement
s : p ( q ( p r ) )
involves three propositions, p, q, and r, each of which may independently be true or false. There are altogether 23 = 8 possible combinations of truth values for p, q, and r, and the truth value for s contains n components statements, there will be 2 n entries needed in the truth table for s.
Step 1. The first n columns of the table are labeled by the component propositional variables.
Further column are constructed for all intermediate combinations of statements,
culminating in the given statement.
Step 2. Under each of the first n headings, we list the 2 n possible n-tuples of truth values of
the component statement s. Each n-tuple is listed on a separate row.
Step 3. For each row we compute, in sequence, all remaining truth values.
Ex. 2 : Compute the truth table of the statement ( p q ) (~ q ~p ). Also ( p q) ( p V q).
Solution : Using steps 1,2 and 3, construct the following table.
p q p q q p q p ( p q) ( q p) p V q ( p q) ( p V q)
T T T F F T T T T
T F F T F F T T F
F T T F T T T T T
F F T T T T T F F
A statement that is true for all possible values of its propositional variables is called a tautology (as in the truth table above).
A statement that is always false is called a contradiction or an absurdity.
A statement that can be either true or false, depending on the truth values of its propositional variables, is called a contingency.
In the example above :
a) The statement ( p q ) ( q p ) is a tautology.
b) The statement p p is an absurdity.
c) The statement ( p q ) ( p V q ) is a contingency.
We have defined a new mathematical structure with two binary operations and one unary operation, [ proposition, , V, ]. It makes no sense to say two propositions are equal; instead we say p and q are logically equivalent, if p q is a tautology. When an equivalence is shown to be a tautology, it means that its two component parts are always either both true or both false, for any values of the proportional variables. Thus the two sides are simply different ways of making the same statement and can be regarded as “equal”. We denote that p is equivalent to q by p q, now we can adopt our properties for operations to say this structure has a property if using equivalent in place of equal gives a true statement, thus we say that they are logically equivalent.
The binary operation V has the commutative property that is p V q q V p. The truth value for
( p V q ) ( q V p ) shows the statement is a tautology.
p q p V q q V p ( p V q ) ( q V p )
T T T T T
T F T T T
F T T T T
F F F F T
OPERATIONS FOR PROPOSITIONS
Theorem 1. The operations for propositions have the following properties.
Commutative properties Distributive properties
1. p V q q V p 5. p V ( q r ) ( p V q ) ( p V r )
2. p q q p 6. p ( q V r ) ( p q ) V ( p r )
Associative properties Idempotent properties
3. p V ( q V r ) ( p V q ) V r 7. p V p p
4. p ( q r ) ( p q ) r 8. p p p
Properties of Negation
9. ~ ( ~ p ) p 10. ~ ( p V q ) ( ~ p ) ( ~ q ) 11. ~ ( p q ) ( ~ p ) V ( ~ q )
Theorem 2 :
a) ( p q ) ((~ p ) V q ) d) ~ ( p q ) ( p ~ q )
b) ( p q ) (~ q ~ p ) e) ~ ( p q ) [ ( p ~ q ) V (q ~ p ) ]
c) ( p q ) [ ( p q ) (q p ) ]
Proof of Theorem 2a using truth table Proof of Theorem 2b using truth table
p q ~p p q ~ p V q ~q ~p p q ~ q ~ p
T T F T T F F T T
T F F F F T F F F
F T T T T F T T T
F F T T T T T T T
Theorem 3 :
a) ~ ( x P( x ) ) x ~ P( x )
b) ~ ( x P( x )) x ~ ( P( x ) )
c) x P( x ) Q( x ) x P( x ) x Q ( x )
d) x P( x ) x Q ( x ) x P( x ) Q ( x )
e) x ( P( x ) V Q ( x ) ) x P( x ) V x Q ( x )
f) x P( x ) Q ( x ) x P( x ) x Q ( x )
g) ( ( x P( x ) ) V ( x Q ( x )) ) x P( x ) V Q ( x ) ( is a tautology )
h) x ( P( x ) Q ( x ) ) x P( x ) x Q ( x ) ( is a tautology )
Theorem 4 : Each of the following is a tautology.
a) p q p f) ~ ( p q ) p
b) p q q g) ( p ( p q ) ) q
c) p p V q h) ( ~ p ( p V q ) q
d) q ( p V q ) i) ( ~ q ( p q ) ) ~ q
e) ~ p ( p q ) j) ( ( p q ) ) ( q r )) ( p r )
Methods of proof
If an implication p q is a tautology, where p and q maybe compound statements involving any number of propositional variables, we say that q logically follows from p. Suppose an implication of the form ( p1 p2 p3 … pn ) q is a tautology, this implication is true regardless of the truth values of any of its components. Then we can say that q logically follows from p1, p2, . . . , pn. When q logically follows from p1, p2, . . . , pn then we write
p1
p2
.
.
.
pn .
q
Virtually all mathematical theorems are composed of implications of the type ( p1 p2 p3 … pn ) q. The pi’s are called the hypotheses, or premise and q is called the conclusion. To prove the theorem means to show that the implication is a tautology.
Note that we are not trying to show that q is true, but only that q will be true if all the Pi’s are true. For this reason , mathematical proofs often begin with the statement “ Supposed that p1, p2, . . . and pn are true ” and conclude with the statement “ Therefore, q is true.” The proof does not show that q is true but simply show that q has to be true if the Pi’s are all true.
Arguments based on tautologies represent correct methods of reasoning. Their validity depends only on the form of the statements involved and not on the truth values of the variables that they contain. Such arguments are called rules of inference. The various steps in the mathematical proof of a theorem must follow from the use of various rules of inference, and a mathematical proof of a theorem must begin with the hypotheses, proceed through various steps, each justified by some rule of inference, and arrive at the conclusion.
Ex. 1: According to Theorem 4 j [ ( p q ) ( q r ) ] ( p r ) is a tautology.
Thus the argument
p q
q r .
p r is universally valid, and so is a rule of inference.
Ex 2: Is the following argument valid ?
If you invest in the stock market, then you will get rich.
If you get rich, then you will be happy. .
If you invest in the stock market, then you will be happy.
Solution : The argument is of the form given in Ex 1, hence the argument is valid, although
the conclusion may be false.
Ex. 3 : The tautology ( p q ) [ ( p q ) (q p ) ] is Theorem 2c, thus both of the following
arguments are valid :
p q
p q q p
( p q ) (q p ) p q
Some mathematical theorems are equivalences, that is they are of the form p q. They usually stated p if and only if q. By Ex 3, the proof of such theorem is logically equivalent with proving both p q and q p, and this almost always the way in which equivalences are proved. We first assume that p is true, and show that q must then be true; next we assume that q is true and show that p must then be true.
A very important rule of inference is :
p
p q
q That is, p is true, and p q is true, so q is true.
Some rules of inference were given Latin names by classical scholars. Theorem 4g is referred to as modus ponens ( the method of asserting ). Theorem 4g is ( p ( p q ) ) q.
Ex 4: Is the following argument valid ?
Smoking is healthy.
If smoking is healthy, then cigarettes are prescribed by physicians .
Cigarettes are prescribed by physicians.
The argument is valid since it is of the form modus ponens. However, the conclusion is false. Observed that the first premise p: smoking is healthy is false. The second premise p q is then true and ( p ( p q ) ), the conjunction of the two premises, is false.
Ex. 5 : If taxes are lowered, then income rises.
Income rises .
Taxes are lowered.
Solution : Let p : taxes are lowered and q : income rises
The argument is of the form p q
q .
p
Assume that p q and q are both true. Now p q may be true with p being false. Then the conclusion p is false. Hence the argument is not valid.
An important proof technique, called an indirect method of proof , follows from the tautology
( p q ) ( ( ~ q ) ( ~ p ) ). This state, as previously mentioned, that an implication is equivalent to its contrapositive. Thus to prove p q indirectly, we assume q is false ( the statement ~ q ) and show that p is then false ( the statement ~ p ).
Ex. 6. Let n be an integer. Prove that if n2 is odd, then n is odd.
Solution : Let p : n2 is odd and q : n is odd. We have to prove that p q is true. Instead, we prove the contrapositive ~ q ~ p. Suppose that n is not odd, so that n is even. Then n = 2k, where k is an integer . Then, n2 = ( 2k )2 = 4k2 = 2 ( 2k2 ), so n2 is even. Thus, we have shown that if n is even, n2 is even, which is the contrapositive of the given statement. Hence, the given statement has been proved.
Another important proof technique is proof by contradiction. This method is based on the tautology ( ( p q ) ( ~ q ) ) ( ~ p ). Thus the rule of inference
p q
~ q .
~ p is valid. Informally, this states that if a
statement p implies a false statement q, then p must be false. This is often applied to the case where q is an absurdity or contradiction, that is, a statement is always false. An example is given by taking q as the contradiction r ( ~ r ). Thus any statement that implies a contradiction must be false. In order to use the proof by contradiction, suppose we wish to show that a statement q logically follows from statements p1, p2, . . ., pn. Assume that ~q is true ( that is, q is false ) as an extra hypothesis p1 p2 . . . pn ( ~ q ) implies a contradiction, then at least one of the statements p1, p2, . . ., pn, ~ q must be false. This means that if all the pi’s are true, then ~ q must be true. Thus q follows from p1, p2, . . ., pn. This is proof by contradiction.
Ex. 7 : Prove there is no rational number p/q whose square is 2. In other words, show that the
2 is rational.
Solution: This statement is good candidate for proof by contradiction., because we can check all possible rational numbers to demonstrate that none had a square equal to 2. Assume ( p/q )2 = 2 for some integers p and q, which have no common factors. Then p2 = 2q2 , so p2 is even. This implies that p is even, since the square of an odd number is odd. Thus, q2 is even, and so q is even. We now have that both p and q are even, and therefore have a common factor 2. This is a contradiction to the assumption. Thus the assumption must be false.
STEPS IN THE PROOF . . . . .
To prove a theorem of the form ( p1 p2 . . . pn ) q, begin with the hypothesis p1, p2, . . . pn and show that some result r1 logically follows. Then using p1, p2, . . . pn, r1 , show that some other statements r2 logically follows. Continue this process, producing intermediate statements r1, r2, . . ., rk called steps in the proof, until we can finally that the conclusion q logically follows from p1, p2, . . . pn, r1, r2, . . . , rk. Each logical step must be justified by some valid proof technique.
Ex. 8 : Prove or disprove the statement that if x and y are real numbers , ( x2 = y2 ) ( x = y ). Solution : The statement can be restated in the form x y R ( x, y ). To prove the statement, we need to provide steps, each of which would be true for all x and y. To disprove the statement, we
need to find one example for which the implication is false.
Since ( – 3 )2 = 32, but – 3 3, the result is false. This example is called a counterexample, and any other counterexample
would do just as well.
Hence, if a statement claims that a property holds for all objects, then to prove it, we must use steps that are valid for all objects of that type. To disprove such statement, we need only to show one counterexample, that is, one particular object for which the claim fails.
State whether the argument given is valid or not. If it is valid, identify the tautology on which it is based.
( 1/62 ) If I drive to work, then I will arrive tired.
I am not tired when I arrive at work . . Valid. (( drive ) tired ) ~ tired ) ( ~ drive )
I do not drive to work.
( 3/62 ) If I drive to work, then I will arrive tired.
I do not drive to work. . Invalid. ( ( d ) t ) ( ~ d ) ( ~ t ) ( ? )
I will not arrive tired.
( 5/62 ) I will become famous or I will not become a writer.
I will become a writer. . Valid ( ( f V ~ w ) w ) f
I will become famous.
( 7/63 ) If I try hard and I have talent, then I will become a musician.
If I become a musician, then I will be happy. .
If I will not be happy, then I did not try hard or I do not have talent.
Valid : [ ( ht m ) ( m hp )] [ ~ hp ~ ht ]
( 13/63 ) Prove that the sum of two odd numbers is even.
Solution : Suppose m and n are odd numbers. Then there exist integers j and k such that
m = 2j + 1 and n = 2k + 1. m + n = 2j + 1 + 2k + 1 = 2j + 2k + 2 = 2 ( j + k + 1 ).
Since j + k + 1 is an integer, m + n is even.
( 15/63 ) Prove that the structure [ even integers, + , * ] is closed with respect to *.
Solution : Suppose m and n are odd numbers. Then there exist integers j and k such that
m = 2j + 1 and n = 2k + 1. m . n = 2j . 2k + 2j + 2k + 1 = 2 ( 2 jk + j + k ) + 1.
Since 2 jk + j + k is an integer, m . n is odd and the system is closed with respect to multiplication.
( 17/63 ) Prove that A = B if and only if A B and B A.
Solution : If A = B, then, clearly, A B and B A. If A B and B A, then A B A
and B must be the same as A.
( 19/63 ) Show that a) A B is necessary and sufficient condition for A B = B.
b) A B is necessary and sufficient condition for A B = A.
Solution : a) If A B, then A B B. But B A B. Hence A B = B. If A B = B, then
since A A B, we have A B.
b) If A B, then A A B. But A B A. Hence A B = A. If A B = A, then
since A B B, we have A B.
( 21/63 ) Prove or disprove : the sum of any 5 consecutive integers is divisible by 5.
Solution : Any 5 consecutive integers can be represented by n, n + 1, n + 2, n + 3, n + 4.
Their sum is 5 n + 10 or 5 ( n + 2 ). This is clearly divisible by 5.
( 23/63 ) Determine if the following is a valid argument. Explain your conclusion.
Prove : x x3 > x2
Proof : x x2 > 0 so x x2 ( x – 1 ) > 0 ( x – 1 ) and x x3 – x2 > 0. Hence x x3 > x2.
Solution : Invalid. Multiplying by x – 1 may or may not preserve the order of the inequality.
( 27/63 ) prove that the sum of two prime numbers, each larger than 2, is not a prime number.
Solution : Let x and y be the prime numbers, each larger than 2. Then x and y are odd and
their sum is even ( Exer. 13 ). The only even prime is 2, so x + y is not prime.
More . . .
and more !
Monday, April 19, 2010
Assignment for Math 211 9:00 - 10:30 daily, to be submitted on April 21, 2010.
1. From an airplane 1,525 m above a horizontal ground, a pilot observes two
villages on the ground to the east whose angles of depression are 8 degrees and 38
minutes and 5 degrees 46 minutes. How far apart are the two villages ?
2. A tree casts a shadow of 12 m when the sun’s angle of elevation is 56.5 degrees.
Determine the height of the tree.
3. Determine the missing parts of a triangle whose hypotenuse is 30 cm if the base
angle is 35.75 degrees.
4. Determine the missing parts of a right triangle whose base is 75 cm and base angle
of 26.5 degrees.
5. An observer notes that the angle of elevation of the top of a 6 m high flag pole
on a level ground is 32.25 degrees. How far is he from the base of the flag pole?
1. From an airplane 1,525 m above a horizontal ground, a pilot observes two
villages on the ground to the east whose angles of depression are 8 degrees and 38
minutes and 5 degrees 46 minutes. How far apart are the two villages ?
2. A tree casts a shadow of 12 m when the sun’s angle of elevation is 56.5 degrees.
Determine the height of the tree.
3. Determine the missing parts of a triangle whose hypotenuse is 30 cm if the base
angle is 35.75 degrees.
4. Determine the missing parts of a right triangle whose base is 75 cm and base angle
of 26.5 degrees.
5. An observer notes that the angle of elevation of the top of a 6 m high flag pole
on a level ground is 32.25 degrees. How far is he from the base of the flag pole?
Friday, April 9, 2010
For CSE 211 students to be submitted on April 12, 2010
Assignment
1. Make a research about the importance of Ethics in IT
2. What is the significance of Ethics for IT in your career ?
For Physics 301 students to be submitted on April 12, 2010
Assignment
1. Convert the following to the specified units.
a. velocity, v = 360 cm/s to ft/s and mi/hr.
b. volume, V = 5,250 cubic meter to cubic yard and cubic feet.
c. Area, A = 450 square meter, to square yard and square feet.
2. A truck moves with a speed of 72 km/hr. Convert this speed in miles per hour, ft/s and m/s.
How many meters will it travel in 45 minutes. How many minutes will it travel 108 km?
Assignment
1. Make a research about the importance of Ethics in IT
2. What is the significance of Ethics for IT in your career ?
For Physics 301 students to be submitted on April 12, 2010
Assignment
1. Convert the following to the specified units.
a. velocity, v = 360 cm/s to ft/s and mi/hr.
b. volume, V = 5,250 cubic meter to cubic yard and cubic feet.
c. Area, A = 450 square meter, to square yard and square feet.
2. A truck moves with a speed of 72 km/hr. Convert this speed in miles per hour, ft/s and m/s.
How many meters will it travel in 45 minutes. How many minutes will it travel 108 km?
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