SAMPLE. Chapter 8 LOGIC EXERCISE 8A.1 EXERCISE 8A.2

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1 Chapter 8 LOGIC EXERCISE 8A.1 1 A proposition is a statement whih may e true or false. A proposition is ineterminate if it oes not have the same answer for all people. a proposition, false proposition, false proposition, true proposition, false e proposition, true f proposition, true g a question, so not a proposition h proposition, true i a question, so not a proposition j proposition, false k proposition, ineterminate l a question, so not a proposition m proposition, ineterminate n proposition, ineterminate o proposition, ineterminate (only true when a transversal rosses two parallel lines) p proposition, false 2 a i :p: not all retangles are parallelograms. p is true. i :m: p 5 is a rational numer. m is true. i :r: 7 is an irrational numer. r is true. i :q: = 12 :q is true. e i :r: = 13 r is true. f i :s: The ifferene etween two o s is true. numers is not always even. g i :t: The prout of onseutive integers is t is true. not always even. h i :u: Not all otuse angles are equal. :u is true. i i :p: Not all trapeziums are parallelograms. :p is true. j i :q: Not all triangles with two equal angles are isoseles. q is true. 3 a x > 5 x<3 y > 8 y>10 4 a i No :r: Kania sore 60% or less. i Yes i No :r: Fari is not at soer pratie. i Yes e i No :r: I i not rink lak tea toay. 5 a x 2f1, 2, 3, 4g x<0, x 2 Z x 2fhorses, sheep, goats, eerg x is a female stuent e x is a female non-stuent EXERCISE 8A.2 SAMLE 1 a = f21, 24, 27g = f2, 4, 6, 8, 10g ' ' = f1, 2, 3, 6, 7, 14, 21, 42g

2 174 Mathematial Stuies SL (3r en), Chapter 8 LOGIC 2 a O M O M M O 3 a If is the truth set of p then = f6, 7, 8g. 4 a If is the truth set of p then = f0, 2, 4, 6, 8g ' ' EXERCISE 8B.1 The truth set of :p is 0 = f9, 10, 11, 12, 13, 14g. 1 a p ^ q: Te is a otor an Shelly is a entist. p ^ q: x is greater than 15 an less than 30. p ^ q: It is winy an it is raining. p ^ q: Kim has rown hair an lue eyes. 2 a p is true an q is true, so p ^ q is true. e The truth set of :p is 0 = f1, 3, 5, 7, 9g. p is true ut q is false (as triangles have three sies), so p ^ q is false. p is false (as 39 > 27) an q is false (as 16 < 23), so p ^ q is false. p is true an q is true, so p ^ q is true. p is false (as 5+8=13) an q is true, so p ^ q is false. 3 a The truth set of p ^ q is \ = f2, 4, 6g EXERCISE 8B.2 1 a p _ q: Tim owns a iyle or a sooter. p _ q: x is a multiple of 2 or a multiple of 5. p _ q: Dana stuies hysis or Chemistry. 2 a p is true an q is true, so p _ q is true. p is false (as a right angle has 90 ± ) ut q is true, so p _ q is true. p is false (as 8 < 5) an q is false (as 5 > 0), so p _ q is false. p is true ut q is false (as the mean of 8 an 14 is 11), so p _ q is true. 3 a p Y q: Meryn will visit Japan or Singapore, ut not oth, next year p Y q: Ann will invite Kate or Tray, ut not oth, to her party. p Y q: x is a fator of 56 or 40, ut not oth. SAMLE

3 4 a a is true an is true, so a Y is false. a is false (as 15 is o) an is true, so a Y is true. a is false (as 4:5 is not an integer) an is false (N oes not inlue negative numers), so a Y is false. a is true ut is false (as (2 8 ) 6 =2 8 6 =2 48 ), so a Y is true. 5 r: Kelly is a goo river an s: Kelly has a goo ar. a :r r ^ s :s ^:r r _ s 6 x: Sergio woul like to go swimming tomorrow an y: Sergio woul like to go owling tomorrow. a :x x ^ y x _ y :(x ^ y) e x Y y 7 a p: hillip likes ieream. q: hillip likes jelly. p ^ q: hillip likes ieream an jelly. p: hillip likes ieream. q: hillip likes jelly. p _:q: hillip likes ie ream or hillip oes not like jelly. p: x is greater than 10. q: x is a prime numer. p ^ q: x is oth greater than 10 an a prime numer. e p: The omputer is on. :p: The omputer is not on. f p: Angela has a wath. q: Angela has a moile phone. :p ^ q: Angela oes not have a wath ut oes have a moile phone. g p: Maya stuie Spanish. q: Maya stuie Frenh. p Y q: Maya stuie one of Spanish or Frenh. h p: I an hear thuner. q: I an hear an aeroplane. p _ q: I an hear thuner or an aeroplane. 8 Sine p Y q is false, p an q are either oth true or oth false. If p an q were oth false, then p _ q woul e false. But p _ q is true, so p is true an q is true. 9 a If is the truth set of p, then = f3, 6, 9, 12, 15, 18g. If is the truth set of q, then = f1, 3, 5, 7, 9, 11, 13, 15, 17, 19g. i The truth set of :q is 0 = f2, 4, 6, 8, 10, 12, 14, 16, 18, 20g. The truth set of p _ q is [ = f1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19g. i The truth set of p ^ q is \ = f3, 9, 15g. Mathematial Stuies SL (3r en), Chapter 8 LOGIC 175 p: Tuan an go to the mountains. q: Tuan an go to the eah. p Y q: Tuan an go to the mountains or to the eah, ut not oth. iv The truth set of p Y q is f1, 5, 6, 7, 11, 12, 13, 17, 18, 19g SAMLE 10 a If is the truth set of p, then = f2, 3, 5, 7, 11g. If is the truth set of q, then = f1, 2, 3, 4, 6, 12g i p ^ q: x is oth prime an a fator of 12. p _ q: x is prime or a fator of 12. i p Y q: x is prime or a fator of 12, ut not oth.

4 176 Mathematial Stuies SL (3r en), Chapter 8 LOGIC i p ^ q has the truth set \ = f2, 3g. p _ q has the truth set [ = f1, 2, 3, 4, 5, 6, 7, 11, 12g. i p Y q has the truth set f1, 4, 5, 6, 7, 11, 12g. 11 a p is false. s is true. q is true an u is true, so q ^ u is true. p is false ut w is true, so p _ w is true. e r is false ut s is true, so r _ s is true. f r is false an s is true, so r ^ s is false. g r is false an s is true, so r Y s is true. h t is false an v is false, so t _ v is false. 12 Let e the truth set of p an e the truth set of q. a p _ q means p or q or oth are true. So we want the region in or or oth, whih is [. :p _ q means :p or q or oth are true. So, p is false, or q is true, or oth. So we want the region in 0 or or oth, whih is 0 [. p Y q means p or q are true ut not oth. So, p is true an q is false, or p is false an q is true. We therefore want the region in ut not, or the region in ut not in. So, ( \ 0 ) [ ( \ 0 ). or p is true, q is true, ut exlue the region where p an q are true. So, ( [ ) \ (( \ ) 0 ) :p ^:q means :p is true an :q is true. So p is false an q is false. We want the region not in an not in, whih is 0 \ 0. SAMLE 13 a The shae region is \, whih is the region in an. So, oth p an q are true, whih is p ^ q. The shae region is the region in or, ut not oth. So, p or q is true, ut not oth, whih is p Y q. The shae region is 0, whih is the region not in. So, p is not true, whih is :p. 14 a The aptain is ol, ut not male. The aptain is ol or male. The aptain is ol. EXERCISE 8C.1 1 a p q :p :p ^ q T T F F T F F F F T T T F F T F p q p Y q :(p Y q) T T F T T F T F F T T F F F F T

5 Mathematial Stuies SL (3r en), Chapter 8 LOGIC 177 p q :p :q :p _:q T T F F F T F F T T F T T F T F F T T T p p _ p T T F F 2 a i p q :p :q :p ^:q T T F F F T F F T F F T T F F F F T T T neither (values in the :p ^:q olumn are neither all true nor all false). i p q p _ q :p (p _ q) _:p T T T F T T F T F T F T T T T F F F T T i p q p Y q p ^ (p Y q) T T F F T F T T F T T F F F F F i p q p ^ q p Y q (p ^ q) ^ (p Y q) T T T F F T F F T F F T F T F F F F F F tautology (values in the (p _ q) _:p olumn are all true). neither (values in the p ^ (p Y q) olumn are neither all true nor all false). logial ontraition (values in the (p ^ q) ^ (p Y q) olumn are all false). 3 a p ^:p is only true if oth p an :p are true at the same time, whih annot our. p :p p ^:p Sine the values in the p ^:p olumn are all false, T F F p ^:p is a logial ontraition. F T F 4 a p :p :(:p) T F T F T F p p ^ p T T F F SAMLE p q :p :p ^ q p _ (:p ^ q) p _ q T T F F T T T F F F T T F T T T T T F F T F F F Sine the truth tale olumns for p an :(:p) are iential, p an :(:p) are logially equivalent. So, :(:p) =p. Sine the truth tale olumns for p an p ^ p are iential, p an p ^ p are logially equivalent. So, p ^ p = p. Sine the truth tale olumns for p _ (:p ^ q) an p _ q are iential, p _ (:p ^ q) an p _ q are logially equivalent. So, p _ (:p ^ q) =p _ q. p q p Y q :(p Y q) :q p Y :q T T F T F T T F T F T F F T T F F F F F F T T T Sine the truth tale olumns for :(p Y q) an p Y :q are iential, :(p Y q) an p Y :q are logially equivalent. So, :(p Y q) =py :q.

6 178 Mathematial Stuies SL (3r en), Chapter 8 LOGIC e p q :p q _:p :(q _:p) :q p _ q :q ^ (p _ q) T T F T F F T F T F F F T T T T F T T T F F T F F F T T F T F F Sine the truth tale olumns for :(q _:p) an :q ^ (p _ q) are iential, :(q _:p) an :q ^ (p _ q) are logially equivalent. So, :(q _:p) =:q ^ (p _ q). f p q :p p _ q :p Y (p _ q) :q p _:q T T F T T F T T F F T T T T F T T T F F F F F T F T T T Sine the truth tale olumns for :p Y (p _ q) an p _:q are iential, :p Y (p _ q) an p _:q are logially equivalent. So, :p Y (p _ q) =p _:q. 5 a p q :p :q :p ^ q p ^:q (:p ^ q) _ (p ^:q) T T F F F F F T F F T F T T F T T F T F T F F T T F F F p Y q is logially equivalent to (:p ^ q) _ (p ^:q). 6 a i p _ q: I like apples or ananas. :(p _ q): I o not like apples or ananas. i :p: I o not like apples. iv :p ^:q: I o not like apples an I o not like ananas. p q p _ q :(p _ q) :p :q :p ^:q T T T F F F F T F T F F T F F T T F T F F F F F T T T T 7 a p q p Y q q ^ (p Y q) (p Y q) _ p T T F F T T F T F T F T T T T F F F F F i p Y q nees only one of p or q true, an not oth. So, 3 6 x 6 7 or x > 2, ut not oth. ) 3 6 x<2 or x>7. From the tale, q ^ (p Y q) nees p false an q true. So, 3 6 x 6 7 is not true, an x > 2 is true. ) x>7 i From the tale, (p Y q) _ p is true when p or q or oth are true. So, 3 6 x 6 7 or x > 2 or oth. ) x > 3 Sine the truth tale olumns for :(p _ q) an :p ^:q are iential, :(p _ q) an :p ^:q are logially equivalent. So, :(p _ q) =:p ^:q. SAMLE 8 a Any tautology has all the values in its truth tale olumn as true, so any two tautologies will have mathing (all true) truth tale olumns. Any logial ontraition has all the values in its truth tale olumn as false, so any two logial ontraitions will have mathing (all false) truth tale olumns. 9 a A logial ontraition has a truth tale olumn of all Fs, so its negation will have all Ts. ) the negation of a logial ontraition is a tautology.