Direct Proof

Overview

The Direct Proof Strategy

A direct proof of "If P, then Q" works by assuming P is true and using definitions, axioms, and previously proven theorems to logically derive Q. This is the most straightforward proof method.

Definition

Direct Proof Template

To prove: If P, then Q.

Proof: 1. Assume P is true. 2. [Logical steps using definitions and theorems...] 3. Therefore Q is true. QED.

Example

Direct Proof: Product of Evens

Theorem: If m and n are even integers, then m × n is even.

Proof: Assume m and n are even integers. By definition, m = 2a and n = 2b for some integers a and b. Then: m × n = (2a)(2b) = 4ab = 2(2ab). Since 2ab is an integer, m × n = 2(2ab) is even by definition. QED.

Example

Direct Proof: Sum of Odd Numbers

Theorem: If m and n are odd integers, then m + n is even.

Proof: Assume m and n are odd. Then m = 2a + 1 and n = 2b + 1 for some integers a, b. m + n = (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1). Since a + b + 1 is an integer, m + n is even. QED.

Example

Direct Proof with Divisibility

Theorem: If a | b and b | c, then a | c.

Proof: Assume a | b and b | c. By definition, b = a × k₁ and c = b × k₂ for integers k₁, k₂. Substituting: c = (a × k₁) × k₂ = a × (k₁ × k₂). Since k₁ × k₂ is an integer, a | c by definition. QED.

Note

When to Use Direct Proof

Direct proof works best when the definitions involved give you a clear algebraic handle. If the hypothesis says "n is even," you can write n = 2k and manipulate from there. If the path from hypothesis to conclusion is not clear, consider proof by contrapositive or contradiction instead.