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138-16-FALL-WEEK4-Oct.3,7-H1.pdf

138-16-FALL-WEEK4-Oct.3,7-H1.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #4 (OCTOBER 3 AND7) PROOFS BY “CASE ANALYSIS” AND “PROOFS COMBINING METHODS”. - Topic: Proofs by “Case Analysis”. This method is used when a complicated proof is broken into into two or more cases that together exhaust all the possibilities. It is most appropriate when the cases can be handled in a relatively straightforward manner. It is usually based on the tautology ( 𝑃𝑃 1 ⇒ 𝑄𝑄 ) ( 𝑃𝑃 2 ⇒ 𝑄𝑄 ) ( 𝑃𝑃 𝑛𝑛 ⇒ 𝑄𝑄 ) � ⇒ � ( 𝑃𝑃 1 ∨ 𝑃𝑃 2 ∨ 𝑃𝑃 𝑛𝑛 ) ⇒ 𝑄𝑄� . Some examples and exercises. Give a proof by case analysis for each of the following statements: 1. ∀𝑛𝑛 ∈ ℕ , 𝑛𝑛 3 + 2 𝑛𝑛 is divisible by 3. 2. ∀𝑛𝑛 ∈ ℕ , 𝑛𝑛 3 + 𝑛𝑛 2 + 5 𝑛𝑛 + 7 is not divisible by 3. 3. If 𝑃𝑃 , 𝑄𝑄 and 𝑅𝑅 are any three propositions, then ( 𝑃𝑃 ⇒ 𝑄𝑄 ) (
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  • Winter '15
  • Math, 2016 Week, Department of Mathematics

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