138-16-FALL-PS2-SOLNS.pdf

138-16-FALL-PS2-SOLNS.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 PROBLEM SET #2. ANSWERS WITH BRIEF EXPLANATIONS. 1. Give a direct proof for the following proposition: ∃𝑛𝑛 ∈ ℕ , ∀𝑥𝑥 ∈ ℝ , ( 𝑛𝑛𝑥𝑥 < 1 + 3 𝑛𝑛 ) ⇒ � 𝑥𝑥 2 + 𝑥𝑥 + 1 𝑥𝑥 2 + 4 < 2017 2016 . Answer: We will show that 𝑛𝑛 = 504 is one of the infinitely many 𝑛𝑛 ∈ ℕ that satisfies the conditions of the problem. First, we should notice that ∀𝑥𝑥 ∈ ℝ , 𝑥𝑥 2 +𝑥𝑥+1 𝑥𝑥 2 +4 > 0 because ∀𝑥𝑥 ∈ ℝ , ( 𝑥𝑥 2 + 𝑥𝑥 + 1 > 0) ( 𝑥𝑥 2 + 4 > 0) . We should also notice that ∀𝑛𝑛 ∈ ℕ + , ∀𝑥𝑥 ∈ ℝ , ( 𝑛𝑛𝑥𝑥 < 1 + 3 𝑛𝑛 ) ( 𝑛𝑛 ( 𝑥𝑥 − 3) < 1) ⇔ �𝑥𝑥 − 3 < 1 𝑛𝑛 . It means that the given proposition is equivalent to ∃𝑛𝑛 ∈ ℕ , ∀𝑥𝑥 ∈ ℝ , �𝑥𝑥 − 3 < 1 𝑛𝑛 � ⇒ � 𝑥𝑥 2 +𝑥𝑥+1 𝑥𝑥 2 +4 < 2017 2016 2 . Notice now that we also have, ∀𝑥𝑥 ∈ ℝ , 𝑥𝑥 2 +𝑥𝑥+1 𝑥𝑥 2 +4 = 1 + 𝑥𝑥−3 𝑥𝑥 2 +4 1 + 𝑥𝑥−3 4 and therefore, taking 𝑛𝑛 = 504 (or any 𝑛𝑛 ≥ 504) , we have, ∀𝑥𝑥 ∈ ℝ , �𝑥𝑥 − 3 < 1 504 � ⇒ � 𝑥𝑥 2 +𝑥𝑥+1 𝑥𝑥 2 +4 < 1 + 1 2016 < 1 + 1 2016 2 = 2017 2016 2 , as required. 2. a) Give a proof by contrapositive for the following proposition: ∀𝑥𝑥 ∈ ℝ + , ∀𝑦𝑦 ∈ ℝ + , ∀𝑧𝑧 ∈ ℝ + , �𝑧𝑧 < 4 𝑥𝑥𝑦𝑦 2 + 𝑥𝑥 2 + 𝑦𝑦 2 � ⇒ � ( 𝑧𝑧 < 𝑥𝑥 ) ( 𝑧𝑧 < 𝑦𝑦 ) . b) Give a proof by contradiction for the following proposition: ∀𝑎𝑎 ∈ ℤ , ∀𝑏𝑏 ∈ ℤ , ∀𝑐𝑐 ∈ ℤ , 𝑎𝑎 ( 𝑎𝑎 − 2) + 𝑏𝑏 ( 𝑏𝑏 − 2) + 𝑐𝑐 ( 𝑐𝑐 − 4) + 𝑎𝑎𝑏𝑏 ( 𝑎𝑎𝑏𝑏 − 2 𝑎𝑎 − 2 𝑏𝑏 + 4) 16. Answers: a) We know that the contrapositive of the conditional predicate �𝑧𝑧 < 4𝑥𝑥𝑥𝑥 2+𝑥𝑥 2 +𝑥𝑥 2 � ⇒ � ( 𝑧𝑧 < 𝑥𝑥 ) ( 𝑧𝑧 < 𝑦𝑦 ) is ( 𝑧𝑧 ≥ 𝑥𝑥 ) ( 𝑧𝑧 ≥ 𝑦𝑦 ) � ⇒ �𝑧𝑧 ≥ 4𝑥𝑥𝑥𝑥 2+𝑥𝑥 2 +𝑥𝑥 2 . So, we will just have to prove the equivalent proposition:
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