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Direct Proof

Definition

A direct proof is a method of proving a statement of the form \(P \implies Q\) by assuming \(P\) is true and deducing that \(Q\) must therefore be true.

Structure

To prove \(P \implies Q\) directly:

  1. Assume \(P\) is true.
  2. Deduce intermediate statements using definitions, axioms, and previously proved theorems.
  3. Conclude that \(Q\) is true.

Why It Matters

Direct proof is the most straightforward proof technique. When it works, it provides the clearest path from hypothesis to conclusion.

Theorem: Sum of Two Even Integers is Even

Statement

For all integers \(a\) and \(b\), if \(a\) is even and \(b\) is even, then \(a + b\) is even.

Proof

Let \(a\) and \(b\) be even integers.

By the definition of even, there exist integers \(k\) and \(m\) such that: $\(a = 2k \quad \text{and} \quad b = 2m\)$

Then: $\(a + b = 2k + 2m = 2(k + m)\)$

Since \(k + m\) is an integer, \(a + b\) is of the form \(2n\) where \(n = k + m\).

Therefore, \(a + b\) is even.

QED

Theorem: Product of Two Odd Integers is Odd

Statement

For all integers \(a\) and \(b\), if \(a\) is odd and \(b\) is odd, then \(ab\) is odd.

Proof

Let \(a\) and \(b\) be odd integers.

By definition, there exist integers \(k\) and \(m\) such that: $\(a = 2k + 1 \quad \text{and} \quad b = 2m + 1\)$

Then: $\(\begin{aligned} ab &= (2k + 1)(2m + 1) \\ &= 4km + 2k + 2m + 1 \\ &= 2(2km + k + m) + 1 \end{aligned}\)$

Since \(2km + k + m\) is an integer, \(ab\) is of the form \(2n + 1\).

Therefore, \(ab\) is odd.

QED

Theorem: If \(n^2\) is Even, Then \(n\) is Even

Statement

For all integers \(n\), if \(n^2\) is even, then \(n\) is even.

Proof

We prove the contrapositive: if \(n\) is odd, then \(n^2\) is odd.

This follows directly from the theorem above (product of two odd integers is odd, and \(n^2 = n \cdot n\)).

Since the contrapositive is true, the original statement is true.

QED

When to Use Direct Proof

Direct proof works best when: - The conclusion follows naturally from the hypothesis - There is a clear chain of implications from \(P\) to \(Q\) - The definitions provide an immediate path forward

When Direct Proof Fails

Direct proof may not work when: - The path from \(P\) to \(Q\) is not obvious - You need to reason about what happens when \(Q\) is false - The statement involves existence or universal quantification in a complex way

In such cases, consider: - Proof by Contradiction - Proof by Contrapositive - Proof by Induction