4.1 Direct Proof and Counter Example I: Introduction

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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

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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

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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

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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.

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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?

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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.

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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.

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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.

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How to Write Proofs

1. Copy the statement.

2. Start your proof with: Proof:

3. Define your variables.

4. Write your proof in complete, grammatically correct sentences.

5. Keep your reader informed.

6. Given a reason for each assertion.

7. Include words or phrases to make the logic clear.

8. Display equations and inequalities.

9. Conclude with .

4.1 Direct Proof and Counter Example I: Introduction

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Example

• Prove the theorem:

The sum of any even integer and any odd integer is odd.

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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.