Discrete Structures

Discrete Structures Chapter 4: Elementary Number Theory and Methods of Proof 4.1 Direct Proof and Counter Example I: Introduction Mathematics, as a science, commenced when first someone, probably a Greek, proved propositions about “any” things or about “some” things without specification of definite particular things. – Alfred North Whitehead, 1861-1947 4.1 Direct Proof and Counter Example I: Introduction

Assumptions We assume that • we know the laws of basic algebra (see Appendix A). • we know the three properties of equality for objects A, B, and C: • A = A • If A = B then B = A • If A = B,B = C, then A = C • there is no integer between 0 and 1 and that the set of integers is closed under addition, subtraction, and multiplication. • most quotients of integers are not integers. 4.1 Direct Proof and Counter Example I: Introduction

Definitions • Even Integer • An integer n is eveniffn equals twice some integer. Symbolically, if n is an integer, then n is even   k  Zs.t. n = 2k. • Odd Integer • An integer n is oddiffn equals twice some integer plus 1. Symbolically, if n is an integer, then n is odd   k  Zs.t. n = 2k + 1. 4.1 Direct Proof and Counter Example I: Introduction

Definitions • Prime Integer • An integer n is primeiffn > 1 for all positive integers r and s, if n = rs, then either r or s equals n Symbolically, if n is an integer, then n is prime   r, s  Z+, if n = rsthen either r = 1 and s = n or s = 1 and r = n. • Composite Integer • An integer n is compositeiffn > 1 and n = rsfor all positive integers r and s with 1 < r < n and 1 < s < n. Symbolically, if n is an integer, then n is composite   r, s  Z+s.t. n = rsand 1 < r < n and 1 < s < n. 4.1 Direct Proof and Counter Example I: Introduction

Example – pg. 161 # 3 • Use the definitions of even, odd, prime, and composite to justify each of your answers. • Assume that r and s are particular integers. • Is 4rs even? • Is 6r + 4s2 + 3 odd? • If r and s are both positive, is r2 + 2rs + s2 composite? 4.1 Direct Proof and Counter Example I: Introduction

Disproof by Counterexample • To disprove a statement of the form “ xD, if P(x) then Q(x),” find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false. Such an x is called a counterexample. 4.1 Direct Proof and Counter Example I: Introduction

Example – pg. 161 # 13 • Disprove the statements by giving a counterexample. • For all integers m and n, if 2m + n is odd then m and n are both odd. 4.1 Direct Proof and Counter Example I: Introduction

Method of Direct Proof • Express the statement to be proved in the form “ x  D, if P(x) then Q(x).” • Start the proof by supposing x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true. (Abbreviated: suppose x  D and P(x).) • Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference. 4.1 Direct Proof and Counter Example I: Introduction

How to Write Proofs • Copy the statement. • Start your proof with: Proof: • Define your variables. • Write your proof in complete, grammatically correct sentences. • Keep your reader informed. • Given a reason for each assertion. • Include words or phrases to make the logic clear. • Display equations and inequalities. • Conclude with . 4.1 Direct Proof and Counter Example I: Introduction

Example • Prove the theorem: The sum of any even integer and any odd integer is odd. 4.1 Direct Proof and Counter Example I: Introduction

Example – pg 162 # 27 • Determine whether the statement is true or false. Justify your answer with a proof or a counterexample as appropriate. Use only the definitions of terms and the assumptions on page 146. • The product of any even integer and any integer is even. 4.1 Direct Proof and Counter Example I: Introduction

Discrete Structures