Direct Proof and Proof by Contrapositive

Transcription

1 Dr. Nahid Sultana October 14, 2012

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3 Consider an implication: p q. Then p q p q T T T T F F F T T F F T

4 Consider an implication: p q. Then p q p q T T T T F F F T T F F T Consider x D, p(x) q(x). It can be expressed as a result or theorem, often written as For x D, if p(x) then q(x) or Let x D. if p(x) then q(x) (1) Thus (1) is true if p(x) q(x) is a true for each x D, and false if p(x) q(x) is false for at least one x D.

5 Consider an implication: p q. Then p q p q T T T T F F F T T F F T Consider x D, p(x) q(x). It can be expressed as a result or theorem, often written as For x D, if p(x) then q(x) or Let x D. if p(x) then q(x) (1) Thus (1) is true if p(x) q(x) is a true for each x D, and false if p(x) q(x) is false for at least one x D. In (1), if q(x) is true for all x D or p(x) is false for all x D, then determining the truth of (1) becomes easier.

6 p q p q T T T T F F F T T F F T If q(x) is true for all x D (regardless the truth value of p(x)). Then x D, p(x) q(x) is a true statement. Such a proof is called a trivial proof.

7 p q p q T T T T F F F T T F F T If q(x) is true for all x D (regardless the truth value of p(x)). Then x D, p(x) q(x) is a true statement. Such a proof is called a trivial proof. Result: Let x R. If x > 0, then x > 0.

8 p q p q T T T T F F F T T F F T If q(x) is true for all x D (regardless the truth value of p(x)). Then x D, p(x) q(x) is a true statement. Such a proof is called a trivial proof. Result: Let x R. If x > 0, then x > 0. If p(x) is false for all x D (regardless the truth value of q(x)). Then x D, p(x) q(x) is a true statement. Such a proof is called a Vacuous proof.

9 p q p q T T T T F F F T T F F T If q(x) is true for all x D (regardless the truth value of p(x)). Then x D, p(x) q(x) is a true statement. Such a proof is called a trivial proof. Result: Let x R. If x > 0, then x > 0. If p(x) is false for all x D (regardless the truth value of q(x)). Then x D, p(x) q(x) is a true statement. Such a proof is called a Vacuous proof. Result: Let x R. If x < 0, then x 5 4.

10 Direct proof of p(x) q(x) for all x D: Assume that p(x) is true for an arbitrary x D. And show that q(x) is true for this x.

11 Direct proof of p(x) q(x) for all x D: Assume that p(x) is true for an arbitrary x D. And show that q(x) is true for this x. Properties of integers: 1. The negative of an integer is an integer. 2. The sum (and difference) of two integers is an integer. 3. The product of two integers is an integer. 4. An integer n is even if n = 2k for some integer k. 5. An integer n is odd if n = 2k + 1 for some integer k.

12 Note: (Writing a proof) 1. Write a proof so that somebody else can read it. 2. Write complete sentences, starting with Proof and ending with (to determine the length of the proof). 3. When introducing a new variable/symbol explain what the symbol is, and what set the variable belongs to. 4. If your equation wraps around, then the equal sign goes at the end of the line, and not before the new line nor both within the text. Or you may choose to center it. For example, within the text like this ab = 4kl + 2k + 2l + 1 = 2(2kl + k + l) + 1. or ab = 4kl + 2k + 2l + 1 = 2(2kl + k + l) + 1

13 Definition: The contrapositive of an implication p q is the implication q p. Example:

14 Definition: The contrapositive of an implication p q is the implication q p. Example: Note: The implication p q and q p are logically equivalent (check the truth tables).

15 Definition: The contrapositive of an implication p q is the implication q p. Example: Note: The implication p q and q p are logically equivalent (check the truth tables). A proof by contrapositive of p q is a direct proof of q p.

16 Definition: The contrapositive of an implication p q is the implication q p. Example: Note: The implication p q and q p are logically equivalent (check the truth tables). A proof by contrapositive of p q is a direct proof of q p. Example: Let x Z. If 5x 11 is even, then x is odd. Example: Let x Z. Then 11x 7 is even if and only if x is odd.