1. – Alfred North Whitehead, 1861-1947
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,
4.1 Direct Proof and Counter Example I: Introduction

2. 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

3. 4.1 Direct Proof and Counter Example I: Introduction
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

4. 4.1 Direct Proof and Counter Example I: Introduction
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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 4.1 Direct Proof and Counter Example I: Introduction
Prove the theorem:
The sum of any even integer and any odd integer is odd.
4.1 Direct Proof and Counter Example I: Introduction

11. 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 sum of any two odd integers is even.
4.1 Direct Proof and Counter Example I: Introduction