4.1 Proofs and Counterexamples

Even Odd Numbers

Find a property that describes each of the following sets• E={…, -4, -2, 0, 2, 4, 6, …}• O={…, -3, -1, 1, 3, 5, …}

Exercise

Use the definitions of even and odd to

a. Justify why -4, 0, 8 are even numbers.b. Justify why 1, 11, −301 are odd numbersc. If a and b are integers, is 6a2b evend. If a and b are integers, is 10a + 8b + 1 odde. Which one of these statement is true

Every integer is even and oddEvery integer is either even or odd

Prime Numbers

If possible write the following positive numbers as a product of positive numbers less than the

given number421818

Justifying Quantified Statement

Determine the truth value of each statement below (Start by writing each statement and its negation

using symbolic logic)1. There exists an even number that is the sum of

two prime numbers

2. Any even number is the sum of two prime numbers

Disproof by Counterexample

Counterexample True

False

Proving Statements

The sum of two even numbers is even

Provide several examples before you make a conjecture whether it is true or false

Indicate how you justify whether the statement is true or false

The sum of two odd numbers is evenProvide several examples before you make a conjecture whether it is true or false

Indicate how you justify whether the statement is true or false

Discuss the truth value of There is a positive integer n such that n2 + 3n + 2 is prime

HINT: a. Analyze different cases b. Look at its negation

DIRECT PROOFS

To prove directly that statement of the form

is true follow these steps : 1. Pick any x (but particular) in the domain and

assume P(x) is true (general-particular)2. Through a series of known true fact conclude

Q(x) is true

Basic Results to master

Make a conjecture about each of the following results and then prove them directly

1. The sum, product, and difference of any two even integers are _______.

2. The sum and difference of any two odd integers are ________.

3. The product of any two odd integers is _____.

4. The product of any even integer and any odd integer is _______.

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

7. The difference of any odd integer minus any even integer is _______.

8. The difference of any even integer minus any odd integer is _______.