1 Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications Kenneth H. Rosen (5 th Edition) Chapter 3 The Foundations: Logic and Proof, Sets and Functions

2 Section 3.1 Methods of Proving Theorems

3 Important Questions When is a mathematical argument correct? What methods can be used to construct mathematical arguments?

4 Terms Theorem A statement that can be shown to be true Proof A demonstration that a theorem is true using a sequence of statement that form an argument. Statements include axioms or postulates along with previously proven theorems.

5 Rules of Inference These statements from propositional logic are used to justify the steps in a proof. The tautology is the basis of the rule of inference called modus ponens, or the law of detachment.

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7 Direct Proofs The implication can be proved by showing that if p is true then q must also be true. This shows that the combination p true and q false never occurs

8 Example Show that if a|b and b|c then a|c. Proof: Assume that a|b and b|c. This means that there exists integer x and y such that b = ax and c = by. But, by substitution we can then say that c = (ax)y = a(xy). But xy is an integer, call it k. Therefore c = ak and by the definition of divisivility, a|c.

9 Indirect Proof Since the implication is equivalent to its contrapositive the original implication can be proven by showing that the contrapositive is true.

10 Example Show that if ab is even then a and b are even. To prove a number is even you must show that it can be written as 2k for some integer k. Since we know that ab is even, ab = 2k for some integer k. But what does that say about a and b? Not much. Consider the contrapositive of the implication: If a and b are not even then ab is not even. That is, if a and b are odd then ab is odd.

11 Example – continued If a number (ab in this case) is odd, we must show that it can be written as 2k+1 for some integer k. But, a and b are odd so there exists integers x and y such that a=2x+1 or b=2y+1. Therefore, ab=(2x+1)(2y+1)=4xy+2x+2y+1= 2(2xy+x+y)+1 Since 2xy+x+y is an integer (call it k) we can write ab as 2k+1 and ab must be odd.

12 Vacuous Proofs If the hypothesis of an implication is false then the implication is true. If it can be shown that p is false, then a proof, called a vacuous proof can be given Example: let P(n) = “if n>1, then n 2 >n” Show that P(0) is true.

13 Trivial Proof If the conclusion of an implication is true then the implication is true. Therefore, if you can show that the conclusion is true, then a proof, called a trivial proof, can be given. Example: Let P(n) = “if a and b are positive integers with a>=b, then a n >=b n ” Show that P(0) is true.

14 Proof by Contradiction Suppose that a contradiction q can be found so that is true, that is true. Then the proposition must be false and p must be true. Example: Prove that is irrational.

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