138-16-FALL-WEEK3-Sept.26,30-H1.pdf

138-16-FALL-WEEK3-Sept.26,30-H1.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #3 (SEPTEMBER 26 AND 30) Part 1. First notions on “Proofs” (Direct and indirect proofs). - Most common types of proofs that we will consider in our course. a) Direct proofs. b) Proofs by contrapositive. c) Proofs by contradiction. d) Proofs by mathematical induction. e) Proofs by case analysis. - Topic: Direct proofs. In most cases, direct proofs are based on the following two valid arguments: a) If 𝑃𝑃 and 𝑃𝑃 ⇒ 𝑄𝑄 , then 𝑄𝑄 (Modus Ponens). That is: �𝑃𝑃 ∧ ( 𝑃𝑃 ⇒ 𝑄𝑄 ) � ⇒ 𝑄𝑄 is a tautology. b) If 𝑃𝑃 ⇒ 𝑄𝑄 and 𝑄𝑄 ⇒ 𝑅𝑅 , then 𝑃𝑃 ⇒ 𝑅𝑅 (Hypothetical Syllogism). That is: ( 𝑃𝑃 ⇒ 𝑄𝑄 ) ( 𝑄𝑄 ⇒ 𝑅𝑅 ) � ⇒ ( 𝑃𝑃 ⇒ 𝑅𝑅 ) is a tautology. - Exercise (some basic examples of direct proofs): Rewrite each of the statements below using variables, quantifiers and logical connectives and then give a direct proof for each of them.
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