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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

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 “ 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

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

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