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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 1.3 Logical Implication Suppose P and Q are two compound statement forms. If the conditional statement form P → Q is a tautology, then we call P → Q an implication and we say P logically implies Q . In this case, whenever P is true, then Q must also be true, so the truth of P “forces” Q to be true. We say that P is a stronger statement form than Q or that Q is weaker than P . Example. Show that the conditional statement forms ( p ∧ q ) → p and p → ( p ∨ q ) are both tautologies, hence p ∧ q is stronger than p and p is stronger than p ∨ q . What can be concluded from these implications? Solution. To show that these statement forms are tautologies, we can just construct their truth tables. p q p ∧ q ( p ∧ q ) → p T T T T T F F T F T F T F F F T p q p ∨ q p → ( p ∨ q ) T T T T T F T T F T T T F F F T Since both statement forms are true for all possible choices of truth values of their statement variables, both are tautologies and hence implications. Since ( p ∧ q ) → p is an implication, then p ∧ q is stronger than p . This means that whenever p ∧ q is true, then p must be true. Similarly, since p → ( p ∨ q ) is an implication, then p is stronger than p ∨ q . So, whenever p is true, then p ∨ q must be true. Example. Show that [( p → q ) ∧ p ] → q is an implication. Solution. To show this statement form is an implication, we can construct its truth table and verify that it is a tautology. p q p → q p ∧ ( p → q ) [ p ∧ ( p → q )] → q T T T T T T F F F T F T T F T F F T F T Since it is true for all choices of truth values of p and q , the statement form is a tautology and hence is an implication. The implication [( p → q ) ∧ p ] → q in the preceding example is called modus ponens , which is Latin for “method of aﬃrming.” This is one of the most widely applied implications both in and out of mathematics. In mathematics, many theorems have the form of a conditional p → q . Suppose we wish to prove that the conclusion of the theorem, q , is true. Since we know that the theorem p → q is true, if we show that the hypothesis of the theorem, p , is true, then the hypothesis ( p → q ) ∧ p in modus ponens is then true. Since this is an implication, we can then conclude that the conclusion q must be true. 25 26 Chapter 1 Propositional Logic For instance, suppose we know (or have proved) the theorem: If the ones digit of an integer is 0, then the integer is divisible by 10. Suppose we then have an integer, say 7,540, whose ones digit is 0. We can then conclude using the theorem and modus ponens that the integer 7,540 is itself divisible by 10. You actually use this line or reasoning all the time without thinking of each of the steps involved....
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- Spring '08
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