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 simply assume P is true and then use logical reasoning to show that Q must also be true. It's straightforward, powerful, and the foundation upon which all other proof techniques are built.
Understand what it means to prove "if P, then Q" and why assuming P is valid
Connect statements logically: P → A → B → ... → Q
Master proofs about integers, divisibility, even/odd, and more
Recognize natural candidates for direct proof vs other techniques
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.
To prove an implication P → Q using direct proof:
The implication P → Q is only false when P is true but Q is false. By showing that P being true forces Q to be true, we prove the implication holds in all cases.
Most direct proofs follow the pattern: Assume P → Apply definition → Algebraic manipulation → Recognize the form of Q → State conclusion. This pattern becomes second nature with practice!
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.
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?
Circular reasoning invalidates the proof. You can only assume P (the hypothesis), never Q (the conclusion). If your proof starts by assuming Q is true, you've made a fatal error.
"n is even" is not useful by itself—you must write n = 2k for some integer k. Definitions give you algebraic objects to work with. Without them, you can't compute anything.
Don't just end with algebra—explicitly state why your result proves Q. For example: "Since n² = 2(2k²), and 2k² is an integer, n² is even by definition. ∎"
If working forward from P seems impossible, consider contrapositive or contradiction. For example, proving "if n² is even, then n is even" is awkward directly but elegant via contrapositive.