Prove P → Q by assuming P and logically deriving Q step by step
Direct proof is the most fundamental proof technique in mathematics. To prove an implication P → Q, you assume P is true and use logical reasoning to show that Q must follow. The standard pattern is: assume P, apply definitions, manipulate algebraically, and arrive at Q.
Direct proof is the foundation upon which all other techniques are built. Contrapositive proves an equivalent statement directly, and contradiction is really a direct proof that ¬P leads to impossibility.
Build a chain of logical implications from hypothesis to conclusion. Each link must follow from the previous one -- no gaps allowed.
Direct proof builds a logical chain from P to Q. Each step follows from the previous one, creating an unbroken path of reasoning.
Assume n is even
In a direct proof, every step must follow logically from the previous one. We start by assuming P is true, then use definitions, algebra, and logic to transform our statements until we reach Q. The chain P → ... → Q forms an unbroken path of valid reasoning.
Key insight: A direct proof is a chain P → A → B → ... → Q. Each step applies a definition, known theorem, or algebraic rule. The chain is only as strong as its weakest link.
Walk through a complete direct proof step by step. At each stage, see exactly which definition or rule justifies the next line.
Walk through complete direct proofs step by step. See how we assume P and logically derive Q.
We want to prove: If m and n are even, then m + n is even.
Key insight: The critical move in most direct proofs is "apply the definition." Translating "n is even" into "n = 2k for some integer k" gives you algebraic objects to manipulate.
Compare how the same theorem can be proved by different techniques. Direct proof works best when the path from P to Q is straightforward.
Direct proof is the most fundamental technique, but it's not always the easiest path. Learn to recognize when to use each method.
| Aspect | Direct | Contrapositive | Contradiction |
|---|---|---|---|
| Strategy | Assume P, derive Q | Assume ¬Q, derive ¬P | Assume ¬P, derive impossibility |
| What you start with | The hypothesis P | Negation of conclusion (¬Q) | Negation of statement (¬P) |
| What you derive | The conclusion Q | Negation of hypothesis (¬P) | A logical impossibility |
| Best used when | Working forward from P is natural | ¬Q gives concrete information | Existence/uniqueness, or all else fails |
| Typical signal | "Q is positive" (something exists, is even, etc.) | "Prove if A then B" where ¬B is simpler | "There is no...", "...is unique", "...is irrational" |
Prove: "If n is even, then n² is even"
What's the best proof technique?
Key insight: Direct proof is the natural first choice when Q is a "positive" statement (something exists, is even, divides, etc.). When the direct path is blocked, that is your signal to try contrapositive or contradiction.