Proofs Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Prove directly: The sum of two even integers is even.

Solution

  1. 1
    Let aa and bb be even integers. By definition, a=2ma = 2m and b=2nb = 2n for some integers m,nm, n.
  2. 2
    Then a+b=2m+2n=2(m+n)a + b = 2m + 2n = 2(m + n).
  3. 3
    Since m+nm + n is an integer, a+b=2(m+n)a + b = 2(m + n) is even by definition.

Answer

a+b=2(m+n) is even.a + b = 2(m+n) \text{ is even.}
A direct proof starts from the hypothesis, applies definitions and algebra, and arrives at the conclusion. Translating 'even' into 2k2k is a standard first step.

About Proofs

A mathematical proof is a rigorous logical argument that demonstrates the truth of a statement beyond doubt, proceeding from accepted axioms and previously proven results through valid inference rules.

Learn more about Proofs →

More Proofs Examples

Example 2 medium

Prove directly: For any integer [formula], if [formula] is odd then [formula] is odd.

Example 3 medium

Prove: The product of two odd integers is odd.

Example 4 medium

Prove directly: if [formula] is even, then [formula] is odd.

← All Proofs Examples Proofs Concept