Direct Proof Math Example 1

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

Example 1

easy
Prove directly: the sum of an even integer and an odd integer is odd.

Solution

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

Answer

a+b=2(m+n)+1 is odd□a+b = 2(m+n)+1 \text{ is odd} \quad \square
A direct proof proceeds from the hypothesis (definitions of even and odd) to the conclusion (the sum is odd) by explicit algebraic steps. The form 2k+12k+1 for some integer kk is the definition of odd.

About Direct Proof

A direct proof establishes a statement P⇒QP \Rightarrow Q by assuming PP is true and using logical steps, definitions, and known theorems to arrive at QQ — the most straightforward proof strategy.

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More Direct Proof Examples

Example 2 medium

Prove directly: for all real [formula], [formula].

Example 3 easy

Prove directly: if [formula], then [formula].

Example 4 medium

Prove directly: for all integers [formula], [formula] is even.

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