Proofs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proofs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A proof is a logically valid argument that establishes a claim from accepted premises.

It is not guessing the answer; it is proving why the answer must be true.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A proof establishes truth beyond doubt by showing the conclusion follows necessarily from axioms and previously proved statements using only valid inference rules.

Common stuck point: Writing "it is obvious" or "clearly" is not a proof step — every gap in reasoning, however small, must be justified explicitly.

Sense of Study hint: Start by rewriting the claim as “if ... then ...” and identify givens and target.

Worked Examples

Example 1

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

Solution

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

Answer

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 2k is a standard first step.

Example 2

medium
Prove directly: For any integer n, if n is odd then n^2 is odd.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Prove: The product of two odd integers is odd.

Example 2

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

These ideas may be useful before you work through the harder examples.

logical statementconditionalproof intuition