138-16-FALL-TT-SOLNS.pdf

138-16-FALL-TT-SOLNS.pdf - UNIVERSITY OF TORONTO DEPARTMENT...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 TERM TEST. ANSWERS WITH BRIEF EXPLANATIONS. 1. a) Write the contrapositive of the conditional proposition �𝑃𝑃 ∨ (~ 𝑄𝑄 ) � ⇒ (~ 𝑅𝑅 ) . b) Write the negation of the quantified proposition ∀𝑥𝑥 ∈ ℝ , ∃𝑛𝑛 ∈ ℕ , ( 𝑛𝑛𝑥𝑥 2 < 1) ( 𝑥𝑥 ≤ 0.001) . c) Let 𝑈𝑈 = , 𝐴𝐴 = �𝑥𝑥 ∈ 𝑈𝑈│𝑥𝑥 > 3 and 𝐵𝐵 = �𝑥𝑥 ∈ 𝑈𝑈│𝑥𝑥 ≥ 4 . List all the elements, if any, of 𝐴𝐴 ∩ 𝐵𝐵 𝐶𝐶 . Answers: a) The contrapositive of �𝑃𝑃 ∨ (~ 𝑄𝑄 ) � ⇒ (~ 𝑅𝑅 ) is the conditional proposition 𝑅𝑅 ⇒ � (~ 𝑃𝑃 ) ∧ 𝑄𝑄� . b) The negation of the quantified proposition ∀𝑥𝑥 ∈ ℝ , ∃𝑛𝑛 ∈ ℕ , ( 𝑛𝑛𝑥𝑥 2 < 1) ( 𝑥𝑥 ≤ 0.001) is the proposition ∃𝑥𝑥 ∈ ℝ , ∀𝑛𝑛 ∈ ℕ , ( 𝑛𝑛𝑥𝑥 2 < 1) ( 𝑥𝑥 > 0.001) . c) The elements of 𝐴𝐴 ∩ 𝐵𝐵 𝐶𝐶 are 2 , 1 , 0 , 1 , 2 and 3 . In effect, 𝐴𝐴 ∩ 𝐵𝐵 𝐶𝐶 = 𝐴𝐴 − 𝐵𝐵 = { 2, 1,0,1,2,3,4,5,6, … } {4,5,6,7,8 … } = { 2, 1,0,1,2,3} . 2. Is the proposition 𝑄𝑄 ⇒ ( 𝑃𝑃 ∨ 𝑅𝑅 ) a valid consequence of the propositions ( 𝑃𝑃 ∧ 𝑄𝑄 ) ⇒ 𝑅𝑅 and ( 𝑄𝑄 ∧ 𝑅𝑅 ) ⇒ 𝑃𝑃 ? Why or why not? Answer: The proposition 𝑄𝑄 ⇒ ( 𝑃𝑃 ∨ 𝑅𝑅 ) is not a valid consequence of the propositions ( 𝑃𝑃 ∧ 𝑄𝑄 ) ⇒ 𝑅𝑅 and ( 𝑄𝑄 ∧ 𝑅𝑅 ) ⇒ 𝑃𝑃 . It is not true that 𝑄𝑄 ⇒ ( 𝑃𝑃 ∨ 𝑅𝑅 ) is true whenever ( 𝑃𝑃 ∧ 𝑄𝑄 ) ⇒ 𝑅𝑅 and ( 𝑄𝑄 ∧ 𝑅𝑅 ) ⇒ 𝑃𝑃 are both true. In effect, consider the case when 𝑃𝑃 , 𝑄𝑄 and 𝑅𝑅 have the truth values F, T and F, respectively. Then, ( 𝑃𝑃 ∧ 𝑄𝑄 ) ⇒ 𝑅𝑅 and ( 𝑄𝑄 ∧ 𝑅𝑅 ) ⇒ 𝑃𝑃 are both true propositions but 𝑄𝑄 ⇒ ( 𝑃𝑃 ∨ 𝑅𝑅 ) is false. 3. Give a direct proof for the following proposition ∀𝑚𝑚 ∈ ℕ , ( 𝑚𝑚 is odd) ⇒ �∃𝑎𝑎 ∈ ℕ , 𝑚𝑚 2 +25 4 + 5𝑚𝑚 2 = 𝑎𝑎 2 . Answer: Clearly, ∀𝑚𝑚 ∈ ℕ , ( 𝑚𝑚 is odd) ( ∃𝑘𝑘 ∈ ℕ , 𝑚𝑚 = 2 𝑘𝑘 + 1) ⇒ �∃𝑘𝑘 ∈ ℕ , 𝑚𝑚 2 +25 4 + 5𝑚𝑚 2 = ( 2𝑘𝑘+1 ) 2 +25 4 + 5 ( 2𝑘𝑘+1 ) 2 . Notice now that ( 2𝑘𝑘+1 ) 2 +25 4 + 5 ( 2𝑘𝑘+1 ) 2 = 4𝑘𝑘 2 +4𝑘𝑘+26 4 + 10𝑘𝑘+5 2 = 4𝑘𝑘 2 +24𝑘𝑘+36 4 = 𝑘𝑘 2 + 6 𝑘𝑘 + 9 = ( 𝑘𝑘 + 3) 2 .
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  • Winter '15
  • Math, Equivalence relation, Proposition, Department of Mathematics

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