138-16-FALL-WEEK1-Sept.12,16-H3.pdf

138-16-FALL-WEEK1-Sept.12,16-H3.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #1 (SEPTEMBER 12 AND 16) Part 3. MORE ON “PROPOSITIONAL CALCULUS” . - Topic: Three important propositions related to the conditional 𝑷𝑷 ⇒ 𝑸𝑸 and commonly used terms. a) The proposition 𝑸𝑸 ⇒ 𝑷𝑷 is usually called “the converse” of the conditional 𝑷𝑷 ⇒ 𝑸𝑸 . b) The proposition (~ 𝑷𝑷 ) (~ 𝑸𝑸 ) is usually called “the inverse” of the conditional 𝑷𝑷 ⇒ 𝑸𝑸 . c) The proposition (~ 𝑸𝑸 ) (~ 𝑷𝑷 ) is usually called “the contrapositive” of the conditional 𝑷𝑷 ⇒ 𝑸𝑸 . - Topic: Two important equivalences related to the conditional 𝑷𝑷 ⇒ 𝑸𝑸 . a) ( 𝑷𝑷 ⇒ 𝑸𝑸 ) ≅ � (~ 𝑸𝑸 ) (~ 𝑷𝑷 ) T he conditional and its contrapositive are equivalent propositions. b) ( 𝑸𝑸 ⇒ 𝑷𝑷 ) ≅ � (~ 𝑷𝑷 ) (~ 𝑸𝑸 ) The converse and the inverse of a conditional are equivalent propositions. - Topic: Valid arguments most commonly used in proofs. a) [( 𝑷𝑷 ⇒ 𝑸𝑸 ) ∧ 𝑷𝑷 ] ⇒ 𝑸𝑸 (Modus Ponens) b) [( 𝑷𝑷 ⇒ 𝑸𝑸 ) (~ 𝑸𝑸 )] (~ 𝑷𝑷 ) (Modus Tollens) c) [( 𝑷𝑷 ⇒ 𝑸𝑸 ) ( 𝑸𝑸 ⇒ 𝑹𝑹 )] ( 𝑷𝑷 ⇒ 𝑹𝑹 ) (Hypothetical Syllogism) - Topic: Most common fallacies found in wrong proofs.
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