Direct Proof Examples in Math

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

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 direct proof starts from assumptions and logically derives the conclusion step by step.

Start from what you know (the hypotheses) and chain logical steps forward until you reach what you want to prove — no detours, no tricks, just forward reasoning.

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: Assumptions plus valid rules should imply the conclusion directly.

Common stuck point: Do not work backwards from what you want to prove — only move forward from hypotheses to conclusion, or you risk circular reasoning.

Sense of Study hint: Begin by expanding definitions in the hypothesis and push implications forward.

Worked Examples

Example 1

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

Solution

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

Answer

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+1 for some integer k is the definition of odd.

Example 2

medium
Prove directly: for all real x, (x-1)(x+1) = x^2 - 1.

Practice Problems

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

Example 1

easy
Prove directly: if x > y > 0, then x^2 > y^2.

Example 2

medium
Prove directly: for all integers n, n^2 + n is even.
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Background Knowledge

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

logical statementconditionalproof intuition