Direct Proof

Example: Proof that the sum of two even integers is even

Consider two arbitrary even integers x and y. The definition of an even integer is: An even integer can be written as 2a where a is an integer. We will show that there exists an integer c such that x + y = 2c. If x + y = 2c, then the sum of x and y is even.

Using the definition of an even integer, the two integers x and y can be rewritten as x = 2a and y = 2b where a and b are integers. Then the sum x + y can be written as 2a + 2b.

Using the distributive property of multiplication, 2a + 2b can be written as 2(a + b). Since a and b are integers, and the set of integers is closed with respect to addition, there exists an integer c = a + b. 2(a + b) can then be rewritten as 2(c) = 2c. Since c is an integer, the expression 2c matches the definition of an even number. QED.

Proof in two column form