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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 1 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 2 Definitions • Even Integer – An integer n is even iff n equals twice some integer. Symbolically, if n is an integer, then n is even k Z s.t. n = 2k. • Odd Integer – An integer n is odd iff n equals twice some integer plus 1. Symbolically, if n is an integer, then n is odd k Z s.t. n = 2k + 1. 4.1 Direct Proof and Counter Example I: Introduction 3 Definitions • Prime Integer – An integer n is prime iff n > 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 = rs then either r = 1 and s = n or s = 1 and r = n . • Composite Integer – An integer n is composite iff n > 1 and n = rs for 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 = rs and 1 < r < n and 1 < s < n. 4.1 Direct Proof and Counter Example I: Introduction 4 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. a. Is 4rs even? b. Is 6r + 4s2 + 3 odd? c. If r and s are both positive, is r2 + 2rs + s2 composite? 4.1 Direct Proof and Counter Example I: Introduction 5 Disproof by Counterexample • To disprove a statement of the form “ xD, 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 6 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 7 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 8 How to Write Proofs 1. 2. 3. 4. 5. 6. 7. 8. 9. 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 9 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 10 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 sum of any two odd integers is even. 4.1 Direct Proof and Counter Example I: Introduction 11