UNIDADES TECNOLOacuteGICAS DE SANTANDER

APUNTES DOCENTES

PROFESOR ALBA VARGAS BUITRAGO

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

ASIGNATURA ALGEBRA SUPERIOR

UNIDADES TECNOLOacuteGICAS DE SANTANDER

Realice la tabla de verdad de las siguientes expresiones indicando si es una contradiccioacuten una tautologiacutea o una proposicioacuten empiacuterica

1 p and q2 (p and q) and r3 not(p rarr notq) and (p and notq)4 (p and q) or (p or notq)5 p and q and r6 not(p and notq) and (p and notq)7 notnot(notp and notq) or (p and notq)8 p or q and r9 not(notp and notq) and (notp and notq)10 p or q and notr11 p and q rarr r12 p and q rarr notr13 p harr not p14 not (p and notq) and (p and notq)15 notnot p or notnot q16 p orq17 p harr q v r18 [ (notp or q) v (p and q)] rarr[ (notp or q) v notp ]19 (p or notq) rarr ( notp rarr notq)20 (p harr notq) v (p or notq)21 (notp and q) or (notp rarr q)22 notq or notp23 (p rarr q and r) harr not(notq v r) v notr24 (notq and r) rarr not(notq v r) v notr25 (p rarr q ) and r 1048774 not(p v r) v notr26 (p rarr q) and (q rarr r) rarr (p and notr)27 notp harr (q and r) or not(notq v r)28 [ (p v not q) rarr (p rarr q)] rarr [(notp rarr q) v notp] v notp29 [ not(p v q) v (p rarr q)] rarr [(notp rarr q) v notp]30 (p rarr q) and (q rarr r) rarr (p and r)31 (p and q rarr r) rarr (p v r ) 32 (p harr q) rarr (notq harr notp)33 (p and q rarr p v notr) rarr not(notq v notr) and r34 (p and q rarr p) rarr (q v r ) and (notp and notr)35 notp rarr (q v r) and nots36 (p harr q) v (p rarr q)37 (p v q) and (notp rarr notq v r)38 (notp and q rarr p v r) harr not(notq v notr) and r39 [ (notp v q) and (r rarr s) ] v nott40 (p and q) v r rarr [ notr rarr (p and q)]41 [ (notp v q ) rarr r ] harr [ (p and notq) v r ]42 (notp or q rarr p and r) harr not(notq v notr) v r43 (p harr q and notr) harr not( notq v notr) v ( r v s )

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

44 (p or notq rarr p and r ) harr [ not(not q v not r) v ( r rarr not q) ]45 (notp harr q) and (p v notp rarr notq v r)46 (notp harr notr and notp) or not(notq rarr notp or r) rarr notr47 (notp harr q v notr) and not(notp harr q v notr)48 p v q harr (p rarr q)49 (p v q) rarr (p harr q)bull50 [ (p v q) rarr (q rarr p) ] v notp51 (p harr q) v ( prarr q)52 (p v q) rarr ( notprarr notq)53 p and notq rarr notp54 (p v q) and (qrarrp)] v not p55 [(ArarrB)(BrarrC)] rarr(ArarrC)56 (p harr notq) and q57 not(notp and q rarr r) rarr (q harr s v t) and (notp and notr)58 notp and q harr p59 p v q harr (p rarr q)60 (notp v q rarr p and r) harr not(notq and notr) and r61 (p v q) harr (notp rarrnotq v r)62 p v q rarr (p harr q)63 notp v (notq and r) rarr (notrharr p)64 (p rarr notq) and (r v notp harr notr)65 ((p v notr rarr notp) and not(notq rarr r) harr notr66 (p rarr q rarr r) rarr p and r67 (p harr q and notr ) harr notnot(notq v notr) v(r v s)68 (notp harr q) larr (p v notp rarr notq v r)69 (p v notqrarr p and r) harr [ not(notq v notr) v (rrarr notq) ]70 (p rarr q) and (q rarr r) rarr (p rarr r)71 (p larr q rarr p) rarr (q v r) and(notq and notr)72 not(p and notq rarr r) rarr not(q harr s v t) and not(not p and not s)73 not(p and notq rarr r) rarr not(q harr notr v q) and not(not p and not notnotp)74 (notq rarr r) v (notr and notp harr notnotr) rarrp and notnotr

Descubra si las siguientes expresiones son EQUIVALENTES es decir si tienen la misma tabla de verdad1 - (p harr q) v (p rarr q)

(p v q) and (notp rarrnotq)2 - p and notq rarr notp

(p harr notq) v q3 - notp v q harr p

p or q

Si p es V y q es F determiacutenese el valor de verdad de las siguientes foacutermulas1- notp larr q2- notp v notp3- notnotp v notnotq4- notq rarr notp5- p rarrnot(p vnotq)6- p v q rarrq

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

7- not(notqrarrp) v (notp rarrq)8- (notp v notq) harr (notp v notq rarr p)9- (p rarr q) v notq rarr notp10- notnotnotp rarr notp

Complete las siguientes frases1 Si ldquop v qrdquo es V y p es F entonces ldquoqrdquo es 2 Si ldquonotq and qrdquo es V entonces ldquoprdquo es 3 Si ldquonotp and notqrdquo es F y ldquoprdquo es F entonces ldquoqrdquo es 4 Si ldquonot (notp v notq)rdquo es V entonces ldquoprdquo es 5 Si ldquop v notqrdquo es F entonces ldquoqrdquo es 6 Si ldquop rarr qrdquo es V y ldquoprdquo es V entonces ldquoqrdquo es 7 Si ldquop rarr qrdquo es V y ldquonotqrdquo es V entonces ldquonotprdquo es 8 Si ldquop harr qrdquo es F y ldquoprdquo es V entonces ldquoqrdquo es 9 Si ldquonotq rarr notprdquo es V y ldquoqrdquo es F entonces ldquoprdquo es 10 Si ldquop and notqrdquo es V entonces ldquop rarr qrdquo es

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

Realice la tabla de verdad de las siguientes expresiones indicando si es una contradiccioacuten una tautologiacutea o una proposicioacuten empiacuterica

1 p and q2 (p and q) and r3 not(p rarr notq) and (p and notq)4 (p and q) or (p or notq)5 p and q and r6 not(p and notq) and (p and notq)7 notnot(notp and notq) or (p and notq)8 p or q and r9 not(notp and notq) and (notp and notq)10 p or q and notr11 p and q rarr r12 p and q rarr notr13 p harr not p14 not (p and notq) and (p and notq)15 notnot p or notnot q16 p orq17 p harr q v r18 [ (notp or q) v (p and q)] rarr[ (notp or q) v notp ]19 (p or notq) rarr ( notp rarr notq)20 (p harr notq) v (p or notq)21 (notp and q) or (notp rarr q)22 notq or notp23 (p rarr q and r) harr not(notq v r) v notr24 (notq and r) rarr not(notq v r) v notr25 (p rarr q ) and r 1048774 not(p v r) v notr26 (p rarr q) and (q rarr r) rarr (p and notr)27 notp harr (q and r) or not(notq v r)28 [ (p v not q) rarr (p rarr q)] rarr [(notp rarr q) v notp] v notp29 [ not(p v q) v (p rarr q)] rarr [(notp rarr q) v notp]30 (p rarr q) and (q rarr r) rarr (p and r)31 (p and q rarr r) rarr (p v r ) 32 (p harr q) rarr (notq harr notp)33 (p and q rarr p v notr) rarr not(notq v notr) and r34 (p and q rarr p) rarr (q v r ) and (notp and notr)35 notp rarr (q v r) and nots36 (p harr q) v (p rarr q)37 (p v q) and (notp rarr notq v r)38 (notp and q rarr p v r) harr not(notq v notr) and r39 [ (notp v q) and (r rarr s) ] v nott40 (p and q) v r rarr [ notr rarr (p and q)]41 [ (notp v q ) rarr r ] harr [ (p and notq) v r ]42 (notp or q rarr p and r) harr not(notq v notr) v r43 (p harr q and notr) harr not( notq v notr) v ( r v s )

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

44 (p or notq rarr p and r ) harr [ not(not q v not r) v ( r rarr not q) ]45 (notp harr q) and (p v notp rarr notq v r)46 (notp harr notr and notp) or not(notq rarr notp or r) rarr notr47 (notp harr q v notr) and not(notp harr q v notr)48 p v q harr (p rarr q)49 (p v q) rarr (p harr q)bull50 [ (p v q) rarr (q rarr p) ] v notp51 (p harr q) v ( prarr q)52 (p v q) rarr ( notprarr notq)53 p and notq rarr notp54 (p v q) and (qrarrp)] v not p55 [(ArarrB)(BrarrC)] rarr(ArarrC)56 (p harr notq) and q57 not(notp and q rarr r) rarr (q harr s v t) and (notp and notr)58 notp and q harr p59 p v q harr (p rarr q)60 (notp v q rarr p and r) harr not(notq and notr) and r61 (p v q) harr (notp rarrnotq v r)62 p v q rarr (p harr q)63 notp v (notq and r) rarr (notrharr p)64 (p rarr notq) and (r v notp harr notr)65 ((p v notr rarr notp) and not(notq rarr r) harr notr66 (p rarr q rarr r) rarr p and r67 (p harr q and notr ) harr notnot(notq v notr) v(r v s)68 (notp harr q) larr (p v notp rarr notq v r)69 (p v notqrarr p and r) harr [ not(notq v notr) v (rrarr notq) ]70 (p rarr q) and (q rarr r) rarr (p rarr r)71 (p larr q rarr p) rarr (q v r) and(notq and notr)72 not(p and notq rarr r) rarr not(q harr s v t) and not(not p and not s)73 not(p and notq rarr r) rarr not(q harr notr v q) and not(not p and not notnotp)74 (notq rarr r) v (notr and notp harr notnotr) rarrp and notnotr

Descubra si las siguientes expresiones son EQUIVALENTES es decir si tienen la misma tabla de verdad1 - (p harr q) v (p rarr q)

(p v q) and (notp rarrnotq)2 - p and notq rarr notp

(p harr notq) v q3 - notp v q harr p

p or q

Si p es V y q es F determiacutenese el valor de verdad de las siguientes foacutermulas1- notp larr q2- notp v notp3- notnotp v notnotq4- notq rarr notp5- p rarrnot(p vnotq)6- p v q rarrq

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

7- not(notqrarrp) v (notp rarrq)8- (notp v notq) harr (notp v notq rarr p)9- (p rarr q) v notq rarr notp10- notnotnotp rarr notp

Complete las siguientes frases1 Si ldquop v qrdquo es V y p es F entonces ldquoqrdquo es 2 Si ldquonotq and qrdquo es V entonces ldquoprdquo es 3 Si ldquonotp and notqrdquo es F y ldquoprdquo es F entonces ldquoqrdquo es 4 Si ldquonot (notp v notq)rdquo es V entonces ldquoprdquo es 5 Si ldquop v notqrdquo es F entonces ldquoqrdquo es 6 Si ldquop rarr qrdquo es V y ldquoprdquo es V entonces ldquoqrdquo es 7 Si ldquop rarr qrdquo es V y ldquonotqrdquo es V entonces ldquonotprdquo es 8 Si ldquop harr qrdquo es F y ldquoprdquo es V entonces ldquoqrdquo es 9 Si ldquonotq rarr notprdquo es V y ldquoqrdquo es F entonces ldquoprdquo es 10 Si ldquop and notqrdquo es V entonces ldquop rarr qrdquo es

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

44 (p or notq rarr p and r ) harr [ not(not q v not r) v ( r rarr not q) ]45 (notp harr q) and (p v notp rarr notq v r)46 (notp harr notr and notp) or not(notq rarr notp or r) rarr notr47 (notp harr q v notr) and not(notp harr q v notr)48 p v q harr (p rarr q)49 (p v q) rarr (p harr q)bull50 [ (p v q) rarr (q rarr p) ] v notp51 (p harr q) v ( prarr q)52 (p v q) rarr ( notprarr notq)53 p and notq rarr notp54 (p v q) and (qrarrp)] v not p55 [(ArarrB)(BrarrC)] rarr(ArarrC)56 (p harr notq) and q57 not(notp and q rarr r) rarr (q harr s v t) and (notp and notr)58 notp and q harr p59 p v q harr (p rarr q)60 (notp v q rarr p and r) harr not(notq and notr) and r61 (p v q) harr (notp rarrnotq v r)62 p v q rarr (p harr q)63 notp v (notq and r) rarr (notrharr p)64 (p rarr notq) and (r v notp harr notr)65 ((p v notr rarr notp) and not(notq rarr r) harr notr66 (p rarr q rarr r) rarr p and r67 (p harr q and notr ) harr notnot(notq v notr) v(r v s)68 (notp harr q) larr (p v notp rarr notq v r)69 (p v notqrarr p and r) harr [ not(notq v notr) v (rrarr notq) ]70 (p rarr q) and (q rarr r) rarr (p rarr r)71 (p larr q rarr p) rarr (q v r) and(notq and notr)72 not(p and notq rarr r) rarr not(q harr s v t) and not(not p and not s)73 not(p and notq rarr r) rarr not(q harr notr v q) and not(not p and not notnotp)74 (notq rarr r) v (notr and notp harr notnotr) rarrp and notnotr

Descubra si las siguientes expresiones son EQUIVALENTES es decir si tienen la misma tabla de verdad1 - (p harr q) v (p rarr q)

(p v q) and (notp rarrnotq)2 - p and notq rarr notp

(p harr notq) v q3 - notp v q harr p

p or q

Si p es V y q es F determiacutenese el valor de verdad de las siguientes foacutermulas1- notp larr q2- notp v notp3- notnotp v notnotq4- notq rarr notp5- p rarrnot(p vnotq)6- p v q rarrq

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

7- not(notqrarrp) v (notp rarrq)8- (notp v notq) harr (notp v notq rarr p)9- (p rarr q) v notq rarr notp10- notnotnotp rarr notp

Complete las siguientes frases1 Si ldquop v qrdquo es V y p es F entonces ldquoqrdquo es 2 Si ldquonotq and qrdquo es V entonces ldquoprdquo es 3 Si ldquonotp and notqrdquo es F y ldquoprdquo es F entonces ldquoqrdquo es 4 Si ldquonot (notp v notq)rdquo es V entonces ldquoprdquo es 5 Si ldquop v notqrdquo es F entonces ldquoqrdquo es 6 Si ldquop rarr qrdquo es V y ldquoprdquo es V entonces ldquoqrdquo es 7 Si ldquop rarr qrdquo es V y ldquonotqrdquo es V entonces ldquonotprdquo es 8 Si ldquop harr qrdquo es F y ldquoprdquo es V entonces ldquoqrdquo es 9 Si ldquonotq rarr notprdquo es V y ldquoqrdquo es F entonces ldquoprdquo es 10 Si ldquop and notqrdquo es V entonces ldquop rarr qrdquo es

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009

UNIDADES TECNOLOacuteGICAS DE SANTANDER

7- not(notqrarrp) v (notp rarrq)8- (notp v notq) harr (notp v notq rarr p)9- (p rarr q) v notq rarr notp10- notnotnotp rarr notp

Complete las siguientes frases1 Si ldquop v qrdquo es V y p es F entonces ldquoqrdquo es 2 Si ldquonotq and qrdquo es V entonces ldquoprdquo es 3 Si ldquonotp and notqrdquo es F y ldquoprdquo es F entonces ldquoqrdquo es 4 Si ldquonot (notp v notq)rdquo es V entonces ldquoprdquo es 5 Si ldquop v notqrdquo es F entonces ldquoqrdquo es 6 Si ldquop rarr qrdquo es V y ldquoprdquo es V entonces ldquoqrdquo es 7 Si ldquop rarr qrdquo es V y ldquonotqrdquo es V entonces ldquonotprdquo es 8 Si ldquop harr qrdquo es F y ldquoprdquo es V entonces ldquoqrdquo es 9 Si ldquonotq rarr notprdquo es V y ldquoqrdquo es F entonces ldquoprdquo es 10 Si ldquop and notqrdquo es V entonces ldquop rarr qrdquo es

DEPARTAMENTO DE CIENCIAS BAacuteSICAS Versioacuten 1 ndash Mayo de 2009