Mam Practica2

 

2

Mbtamât`db Bpi`dbhb b ib Muit`mah`b Zrâdt`db # >7 Iôo`db Mbtamât`db @.

@eh` @eh`qu qua a ai vb vbik ikrr ha va varh rhbh bh ha db dbhb hb ue ueb b ha ib ibs s pr prkp kpks ks`d `d`k `kea eas s s`ou`aetas7

p7

> as ue e÷mark `mpbr. S (p)= Cbisk 

q7

? as ue e÷mark pr`mk. S (q)=  Sarhbhark 

r7

1/? ek as ue e÷mark aetark. S (q)=  Cbisk 

s7

Tbetk Hkm`eok as ue pbís. S (s)=  Sarhbhark 

t7

Ib t`arrb o`rb birahahkr hai ski. S (t)=  Sarhbhark 

@@.

B dket`eubd`ôe, sa prasaetbe ueb sar`a ha prkpks`d`keas. @eh`qua dke ueb D ibs qua ske dkmpuastbs y dke ueb T ibs qua ske s`mpias.

2. T  Dbriks as ue ekmnra ha parskeb. >. D  Ai `ev`arek as ueb astbd`ô astbd`ôe e hai bòk y Gube Zbnik H Hubrta ubrta hasdunr`ô b Bmàr`db Saeazuaib eazuaib as uek ha iks p pbísas bísas ha mbykr prkhud prkhudd`ôe d`ôe patrkiarb. 4. T  Sa 8. D 4+> =1 y 1 as ue e÷mark pr`mk. 1. D 3 as ue e÷mark pks`t`vk s` y skik s` as mbykr qua dark.

@@@.

Hbhks p7 Ai `ev`arek as crík< q7 H`beb as astuh`ksb, r7 Ib sbibmbehrb  as ue be`mbi vartanrbhk vartanrbhk.. Asdr`nb ae iaeoubga dkikq dkikqu`bi7 u`bi7

cr`k..  b) ~p7 Ai `ev`arek ek as cr`k n) ~q7 H`beb ek as astuh`ksb.

d) ~r7 Ib sbibmbehrb ek as ue be`mbi vartanrbhk.  h) p U ~q7 Ai `ev`arek as cr`k y H`beb ek as astuh`ksb.

a) ~q ⋂  ~r7 H`beb ek as astuh`ksb k ib sbibmbehrb ek as ue be`mbi vartanrbhk. c) (p U q) v r7 Ai `ev`arek as cr`k y H`beb as astuh`ksb k ib sbibmbehrb as ue be`mbi vartanrbhk.

o) (p  ⋂ ~q]) U ~r ~r77 (Ai `ev`arek as cr`k k H`beb ek as astuh`ksb) y ib sbibmbehr sbibmbehrb b ek as ue be`mbi vartanrbh vartanrbhk. k.

 

>

f) ~r U (p

q)7 Ib sbibmbehrb ek as ue be`mbi vartanrbhk y (Ai `ev`arek as cr`k k H`beb

as astuh`ksb).

`) (q U ~r) (~p U q)7 (H`beb as astuh`ksb y ib sbibmbehrb ek as ue be`mbi vartanrbhk) (Ai `ev`arek ek as cr`k y H`beb as astuh`ksb.   g) ~r

 ~q7 Ib sbibmbehrb ek as ue be`mbi vartanrbhk s` y skik s` H`beb ek as

⟿

astuh`ksb. l) q i) ~p

↞

 ~r7 T` H`beb as astuh`ksb, aetkedas ib sbibmbehrb ek as ue be`mbi vartanrbhk. vartanrbhk.  \(p U q)

↞

↘

  (~q U r)7 T` ai `ev`arek ek as cr`k aetkedas (ai `ev`arek as cr`k

k H`beb as astuh`ksb) s` y skik s` (H`beb ek as astuh`ksb y ib Tbibmbehrb as ue be`mbi vartanrbhk. @S. Axprasa ae iaeoubga mbtamât`dk dbhb ueb ha ibs s`ou`aetas prkpks`d`keas7 b) p U q Ai `ev`arek as crík y H`beb as astuh`ksb. n) (~r U q) v ~p (Ib sbibmbehrb ek as ue be`mbi vartanrbhk y H`beb ek as `etai`oaeta) k ai `ev`arek ek as crík.   d) p v q  Ai `ev`arek as crík k ib sbibmbehrb as ue be`mbi vartanrbhk.  h) ~qU (p v r) H`beb ek as astuh`ksb y (ai `ev`arek as crík k ib sbibmbehrb as ue be`mbi vartanrbh vartanrbhk). k). a) (q U p) U (r v ~p) (H`beb as astuh`ksb y ai `ev`arek as crík) y (ib sbibmbehrb as ue be`mbi vartanrbhk k ai `ev`arek ek as crík). 

r T` ai `ev`arek as crík y H`beb as astuh`ksb aetkedas c) p aetkedas ib sbibmbehrb as ue be`mbi vartanrbhk. o) ~r

q T` ib sbibmbehrb ek as ue be`mbi vartanrbhk s` y skik s` H`beb as

astuh`ksb.  astuh`ksb. f)  (~p U q) r Ai `ev`arek ek as crík y H`beb as astuh`ksb y skik s` ib sbibmbehrb as ue be`mbi vartanrbhk. `)  (~p v r) ~r  T` ai `ev`arek ek as crík k ib sbibmbehrb as ue be`mbi vartanrbhk, aetkedas H`beb ek as astuh`ksb. 

S. Dkestruya dkrradtbmaeta ib tbnib ha varhbh ha dbhb ueb ha ibs prkpks`d`keas s`ou`aetas. 2) p ¬q p

q

~q

pU~ q

 

4 S S C C

S C S C

C C S S

>)

(p q)  r

p

q

r

C C C C

(pUq

(pUq) v r

) S S S S C C C C

S S C C S S C C

S C S C S C S C

S S C C C C C C

S S S C S C S C

(¬p  ¬q) ¬r

4)

p

q S S S S C C C C

r S S C C S S C C

S C S C S C S C

¬p C C C C S S S S

¬q C C S S C C S S

¬r C S C S C S C S

(¬p¬q) C C S S S S S S

  ¬p  ¬q

8)

p

q

S S C C

S C S C

~ p C C S S

~ q C S C S

~p v ~q C S S S

(p  q)   (p  r)

1)

p

q

r

(pUq

(pUr)

(pUq) v (pUr)

) S

S

S

S

S

S

S S

S C

C S

S C

C S

S S

(¬p¬q)  ¬r C C C S C S C S

 

8 S C C C C

C S S C C

C C C C C

C C C C C

C C C C C

(¬p  ¬q)   (¬p  s)

3)

p

q

S S S S C C C C ?)

C S C S C

s S S C C S S C C

~p

~q

C C C C S S S S

C C S S C C S S

S C S C S C S C

(~p v ~  ~q q)

(~p v s  s))

C C S S S S S S

S C S C S S S S

¬p   \q   (r  s)] p

q

r

s

~p

(r v s  s))

\q U (r v s)]

~p U \q U (r v s)]

S S S S S S S S C C C C C C C C 6)

(~p v ~q) U (~p v s)

S S S S C C C C S S S S C C C C

S S C C S S C C S S C C S S C C

S C S C S C S C S C S C S C S C

C C C C C C C C S S S S S S S S

S S S C S S S C S S S C S S S C

S S S C C C C C S S S C C C C C

¬p  s p

~p

s S

S

C

~p ↞s S

S

C

C

S

C C C C C C C C S S S C C C C C

C C S C S S S S

 

1 C

S

S

S

C

C

S

C

(p  ¬r)  q

0)

p

q S S S S C C C C

¬r

r S S C C S S C C

S C S C S C S C

(pU¬r)  q

(pU¬r )

C S C S C S C S

C S C S C C C C

C S S C C C S S

2:) \(¬p  ¬r)  (q  ¬p)]  (¬r  ¬p)

p

q

r

¬p

¬q

S S S S C C C C

S S C C S S C C

S C S C S C S C

C C C C S S S S

C C S S C C S S

¬r C S C S C S C S







(¬p ¬r)

(q ¬p)

(¬r ¬q)

C C C C C S C S

C C S S S S S S

C C C S C C C S



\(¬p ¬r)  (q¬p)] S S C S C C C S





\(¬p ¬r) 







S S C S C C C S

  @S @S.. Hbhbs ibs prkpks`d`keas s`ou`aetas rabi`da ib tbnib ha varhbh ha dbhb  ueb ha aiib a `eh`qua dubias ske tbutkikoíb, dketrbh`dd`keas k  (Fbdaribs ae fkgbs suaitbs).  suaitbs).   dket`eoaed`bs. (Fbdaribs dket`eoaed`bs.

2. p

q

\p v (p ↞ q)] ↞ q (p ↞

\p v( v(p ↞q ↞q)]

\p v( v(p ↞q ↞q)] ↞ q

q) S S C

S C S

S C S

S S S

S C S

 C

C

S

S

C



 (q ¬p)]  (¬r ¬q)

Dket`eoaed`b

 

3

>. p

q

(p ↞ q) ↘ (p U ~q) ~

(p ↞

q

q)

(p U ~ q)

(p ↞ q ) ↘ (p U ~  q)

S S

S C

C S

S C

S S

S C

C C

S C

C S

S S

C S

C S

4. p

q

S S C C

S C S C

8. p S S C C

q S C S C

1.

~p ↞ (p v q) ~p

(p v  q )

C C S S

~p ↞ (p v q)

S S S C

S S S C

Dket`eoaed`b

(~p ↞ ~q) U (~p U q) ~

~

(~p ↞

p

q

~q )

C C S S

C S C S

(~p U q)

(~p ↞ ~q )U (~p

U q) S S C S

S C S S

S C C S

 D  k e  t   `     e   o   a  e  d  `      b

\q U (~p ↞ q)] ↞ q

p

q

~p

S S C C

S C S C

C C S S

S.

Dket`eoaed`b

(~p ↞  q ) \q U (~p ↞  q )] S S S C

S S S S

\q U (~p ↞  q )] ↞ q S C S C

Dket`eoaed`b

T` B= {:, 2, >, 4, 8, 1} hatarm`ea ai vbikr ha varhbh ha dbhb ueb ha ibs prkpks`d`keas s`ou`aetas. Gust`c`qua su raspuastb.

2.   x  B< 8 x 9 >: Cbisk pkrqua 8(1) = >:

 

? >.    x  B< x > - 2 ≡ : Sarhbhark

4.    x  B< 4x + 3 5 8 x Sarhbhark

8.    x  B< x > + >x = : Sarhbhark

1.    x  B< 8x + 2 5 >: Cbisk

3.    x  B< x > - >x 5 1 Cbisk

  S@.

^asuaiva dkrradtbmaeta iks s`ou`aetas prkniambs iôo`dks7

2. Qe pbstkr t`aea qua pbsbr ue iknk, ueb dbnrb y ueb iadfuob b ib ktrb kr`iib ha ue

rík, h`spkea ha ueb nbrdb ae ib qua sôik dbnae ài y ueb ha ibs ktrbs tras dksbs. T` ai iknk sa quahb skik dke ib dbnrb sa ib dkma, s` ib dbnrb sa quahb skib dke ib iadfuob sa ib dkma, »Dômk hana fbdarik;. Ai pbstkr pbsb pr`mark ib dbnrb, ib hagb ae ib ktrb kr`iib y raorasb b pkr ai iknk. Bi druzbr, hagb bi iknk y vuaiva dke ib dbnrb, hagb ib dbnrb y druzb dke ib iadfuob. Zkr ÷it`mk, hagb ib iadfuob dke ai iknk y raorasb b pkr ib dbnrb. >. Qe fkmnra t`aea ^H$2::.:: dke iks dubias hana dkmprbr 2:: be`mbias ha tras

vbr`ahbhas. Tkik qua ai prad`k as ai s`ou`aeta, vbdb b ^H$2:.::, df`vk b ^H$ 1.:: y pkiiks b :.1: daetbvks. Dkmk ik fbríb ai fkmnra pbrb dke iks 2:: pasks dkmprbr 2:: be`mbias. Ai fkmnra hana dkmprbr 2 darhk, 1 vbdbs y 08 pkiiks. 4. Qe ksk dbm`eb 2: Lm. Fbd`b ai sur, 2: fbd`b ai asta y 2: fbd`b ai ekrta, vkiv vkiv`a `aeh ehk k bi puet puetk k hai hai qua qua pbrt` pbrt`ô. ô. »H »Haa duâi duâi dkik dkikrr as ai ksk; ksk; Gust Gust`c `c`d `dbb tu raspuastb. Ai ksk as ha dkikr nibedk, pkr sar ue ksk pkibr. Pb qua iks ÷e`dks iuobras hkeha sa dumpia ib dkeh`d`ôe ha raorasbr bi puetk ha pbrt`hb ske ai Zkik Ekrta y dubiqu`ar puetk s`tubhk b 2:lm bi ekrta ha iks pbrbiaiks qua m`hbe 2:lm ha asta vkivaramks bi puetk ha pbrt`hb.

  S@@.

Zruanb tu `etai`oaed`b oaearbi y dbpbd`hbh ha dkedaptubi`zbr y bpi`dbr ai rbzkebm`aetk s`stamât`dk b euavks prkniambs. H`nugb ib c`dfb qua s`oua ae astb sar`a. 

 

6

 

S@@@.

^asuaiva dkrradtbmaeta iks s`ou`aetas prkniambs iôo`dks7

8. Qe fkmnra astâ bi pr`ed`p`k ha ue ibrok pbs`iik qua t`aea tras `etarruptkras, bi

c`ebi fby ueb fbn`tbd`ôe dke ib puartb darrbhb. Qek ha astks tras `etarruptkras

 

0 aed`aeha ib iuz ha asb fbn`tbd`ôe, qua astâ `e`d`bimaeta bpbobhb. »Dômk ik f`zk pbrb dkekdar qua `etarruptkr aed`aeha ib iuz radkrr`aehk ueb skib vaz ai trbyadtk hai pbs`iik; Z`stb7 Ai fkmnra t`aea ueb i`etareb. @iustra dke ue h`nugk. Cbisk, ai fkmnra aed`aeha ai saouehk `etarruptkr, asparb 2: m`eutks y iuaok ik bpbob y aed`aeha ai 4ar `etarruptkr. Ha `emah`btk aetrb ib fbn`tbd`ôe7 T` ai nkmn`iik astb aedaeh`hk ai `etarruptkr dkrradtk as ai 4rk (yb qua sa quahô aedaeh`hk), s` ai nkmn`iik astâ bpbobhk park dbi`aeta aetkedas ai `etarruptkr dkrradtk as ai >hk (yb qua sa aedaeh`hk pkr ue t`ampk y iuaok sa bpboô) y pkr uit`mk s` ai nkmn`iik astâ bpbobhk y cr`k ai `etarruptkr dkrradtk as ai 2rk (yb qua euedb cua aedaeh`hk). Aedaehar, asparbr 2: m`eutks y bpbobr.

 

Aedaehar y hagbr aedaeh`hk.

1. Xras bm`oks dke h`c`duitbhas adkeôm`dbs dkmpbrtae ue dbcà qua ias duastb 4:

pasks, pkr ik qua dbhb uek pkea 2:. Dubehk vbe b pbobr p`hae ue hasduaetk y ai huaòk ias ranbgb 1 pasks tkmbehk dbhb uek ue pask y hagbehk hks ae ue ckehk dkm÷e. Mâs tbrha fbdae duaetbs y h`dae7 Dbhb uek fb pbobhk 0 pasks bsí qua famks pbobhk 0x4=>? pasks qua dke iks hks hai ckehk fbdae >0 »Hôeha astâ ai pask qua cbitb; Ae rabi`hbh, ek cbitb e`eo÷e pask, as qua skibmaeta fby ue arrkr ha dâiduik, iks hks pasks hai ckehk ek sa sumbe b ik pbobhk, s`ek qua sa rastbe, ib kparbd`ôe saríb 0x4=>? pasks pbobhbs >?->=>1 obstbhbs.

  3. Qe aeduastbhkr sa h`r`oa b ueb dbsb hkeha as btaeh`hk pkr ueb mugar7 »Dbet`hbh ha f`gks; Xras h`da aiib. »Ahbhas;, Ai prkhudtk ha ibs ahbhas as 43 y ib sumb as `oubi bi e÷mark ha ib dbsb, raspkeha. Ai aeduastbhkr sa vb park bi rbtk vuaiva y ia h`da b ib mugar qua iks hbtks qua ia h`k ek ske suc`d`aetas< ib mugar p`aesb y ia h`da h`da77 t`a t`aea ea rbzô rbzôe, e, ib mb mbyk ykrr astu astuh` h`bb p` p`be bek. k. As Astk tk as suc` suc`d` d`ae aeta ta pbrb pbrb qua qua ai aeduastbhkrr sapb ibs ahbhas ha iks f`gks. »Duâias ske; aeduastbhk Ai aeduastbhkr praouetb ibs ahbhas y bi kntaear dkmk raspuastb qua ai prkhudtk ha astbs as 43 y su sumb ai e÷mark ha ib dbsb, m`rb ai e÷mark ha astb, qua ek sbnamks. Ai aeduastbhkr hasdkmpkea ai 43 ae sus cbdtkr`bias y rabi`zb ibs s`ou`aetas dkmn`ebd`keas ha ahbhas (tkhbs ibs pks`nias)7

2-2-43 | 2->-26 | 2-4-2> | 2-8-0 2-3-3 | >->-0 | >-4-3 | 4-4-8

 

2: Tkik quahb sbnar duâi ha astbs dkmn`ebd`keas ha ahbhas sumbe ai e÷mark ha ib dbsb. Ai aeduastbhkr sa hb duaetb ha qua ia cbitb bio÷e hbtk, skik puaha prkhud`rsa pkrqua fby hks dkmn`ebd`keas qua sumbe `oubi7

2+3+3=24 >+>+0=24 Bi raorasbr y sbnar qua ib mbykr astuh`b p`bek, hahuda qua skik fby ueb mbykr, ek hks, pkr ik qua ibs ahbhas sarâe >, > y 0 bòks.

@V.

Taiadd`keb ib raspuastb dkrradtb7