Prove P → Q by assuming P and logically deriving Q step by step
To prove “if P then Q,” assume P and reason forward, one justified step at a time, until Q drops out. Every other proof technique is built on this one.
Pick an example, then step through or auto-play the chain from hypothesis to conclusion. Notice what justifies each link.
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.
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.
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.
Pick a statement and press Next Step to walk the proof line by line. Watch for the moment a definition turns words into algebra.
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.
Read each statement and choose the best technique: direct, contrapositive, or contradiction. Open the decision flowchart if you get stuck.
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?
Compare how the same theorem can be proved by different techniques. Direct proof works best when the path from P to Q is straightforward.
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.
Spot a mistake or have an idea?