Lecture 5(Wedneday) - contrapositive and negation Another...

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Unformatted text preview: contrapositive and negation Another symbol, X, toggles the truth value of a statement. When we toggle and reverse an implication, we get its contrapositive. Compare the meanings of: V P ( )A ( ) V P X ( )AX ( x E; F x L x x E; L x F x ) What information does each form give you in each of the four following cases: 1. When x P F ? 2. When x TP F ? 3. When x P L? 4. When x TP L? What would a counter-example to each form be? slide 12 numerical example De ne P (n) : n is a multiple of 4, and Q(n) : n2 is a multiple of 4, and consider V PN n ;P n ( ) A Q(n) What do the implication, converse, and contrapositive each tell you when n is a multiple of 4 is not a multiple of 4 is a multiple of 4 is not a multiple of 4 n n 2 n 2 Which do you believe, and why? slide 13 \natural" language Here are some ways of expressing implication, P each case? If nominated, I will not stand. If you think I'm lying, then you're a liar! Whenever I hear that song, I think about icecream. Dierentiability is sucient for continuity. Matching ngerprints and a motive are enough for guilt. You can't stay enrolled in CSC165 without a pulse. Successful programming requires skill. I'll go only if you insist. Don't knock it unless you've tried it. slide 14 A Q , in English. What's P and what's Q, in vacuous truth We've already separated implication from quanti cation, so we can make sense of P x ( ) A Q(x) It's true, except when P (x) is true and Q(x) is false. In particular, an implication is always true when the antecedent is false. For example, if your eyes wander to the consequent in V PR x ;x 2 2 x +2=0Ax>x+5 . . . you could jump to the conclusion that the implication is false. Vacuous truth works because there are no counterexamples. Another way of thinking about this is that the empty set is a subset of every other set. All employees earning over 80 trillion dollars are female. All employees earning over 80 trillion dollars are male. All employees earning over 80 trillion dollars have mauve eyeballs and breathe ammonia. slide 15 ...
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This note was uploaded on 10/13/2011 for the course COMPUTER S CSC 165 taught by Professor Dannyheap during the Fall '10 term at University of Toronto- Toronto.

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Lecture 5(Wedneday) - contrapositive and negation Another...

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