WUCT121 Discrete Mathematics Logic Tutorial Exercises Solutions

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Section 1: LogicQuestion1(i) If x = 3 , then x < 2 .(a) Statement(b) False(c) x = 3 ⇒ x < 2(ii) If x = 0 or x = 1, then x = x(a) Statement(b) True(c)( x = 0 ∨ x = 1) ⇒ x = x(iii) There exists a natural number x for which(a) Statement(b) False(iv) If(a)(b)22x ∈and x > 0 , then if x > 1 then x > 1..StatementTrue(c) ( x ∈∧ x > 0) ⇒ ( x > 1⇒x > 1).2x = −2x.(v) xy = 5 implies that either x = 1 and y = 5 or x = 5 and y = 1.(a) Statement(b) False. Consider x = −1and y = −5or x = −5and y = −1.(c) xy = 5 ⇒ (( x = 1 ∧ y = 5) ∨ ( x = 5 ∧ y = 1))(vi) xy = 0 implies x = 0 or y = 0 .(vii)(a) Statement(b) True(c) xy = 0 ⇒ x = 0 ∨ y = 0xy = yx .(a) Statement(b) True(viii) There is a unique even prime number.(a) Statement(b) True, x = 2.WUCT121 Logic Tutorial Exercises Solutions 2
Question2(a) If x is odd and y is odd then x + y is even.p: x is odd. q: y is odd. r: x + y is even.Form: p ∧ q ⇒ r .(b)(c)(d)It is not both raining and hot.p: It is raining. q: It is hot~ p ∧ q , alternatively ~ p ∨ ~ qForm: ( )It is neither raining nor hot.p: It is raining. q: It is hotForm: p ~ q~ ∧ , alternatively ( p ∨ q)It is raining but it is hot.p: It is raining. q: It is hot.Form: p ∧ q .~ .(e) −1 ≤ x ≤ 2 .p : −1< x,q : −1= x,r : x < 2, s : x = 2 .Form: ( p ∨ q)∧ ( r ∨ s).Question3 :(a) P ∨ Q : Mathematics is easy or I do not need to study.(b)(c)(d)(e)(f)(g)P ∧ Q : Mathematics is easy and I do not need to study~ Q : I need to study.~ ~ Q : I do not need to study.~ P : Mathematics is not easy.~ P ∧ Q : Mathematics is not easy and I do not need to study.P ⇒ Q : If Mathematics is easy, then I do not need to studyQuestion4(a) The truth tables for ( ~ p ∨ q) ∧ q and ( p ∧ q) ∨ q~ .p q (~p ∨ q) ∧ q (~p ∧ q) ∨ qT T F T T F F TT F F F F F F FF T T T T T T TF F T T F T F FStep: 1 2 3* 1 2 3*The tables are the same(b) The truth tables for ( ~ p ∨ q) ∧ p and ( p ∧ q) ∨ p~ .p q (~p ∨ q) ∧ p (~p ∧ q) ∨ pT T F T T F F TT F F F F F F TF T T T F T T TF F T T F T F FStep: 1 2 3* 1 2 3*The tables are not the same. The student’s guess is falseWUCT121 Logic Tutorial Exercises Solutions 3
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WUCT121 Discrete Mathematics Logic Tutorial Exercises Solutions

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