Proofs Math Example 4

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

Example 4

medium
Prove directly: if nn is even, then n+1n+1 is odd.

Solution

  1. 1
    Assume nn is even, so n=2kn = 2k for some integer kk.
  2. 2
    Then n+1=2k+1n+1 = 2k+1, which has the form of an odd integer. Therefore n+1n+1 is odd.

Answer

n+1=2k+1 is odd.n+1 = 2k+1 \text{ is odd.}
Direct proofs often begin by translating the hypothesis into its algebraic definition. Once n=2kn = 2k is written down, the conclusion follows immediately by adding 1.

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.

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More Proofs Examples

Example 1 easy

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

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.

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