Picking a proof technique is usually less mysterious than it looks. Most of the time, one heuristic will get you the right first attempt: if the hypothesis is a clean, positive property, reach for direct proof before anything else. Out of the four main techniques in your toolkit — direct, contrapositive, contradiction, induction — direct is by far the most common and by far the cheapest to abandon if it fails. So it earns the first look.
This article is about recognising the signal: what does a "clean hypothesis" look like, and why does that signal reliably point to direct proof?
What counts as a "clean" hypothesis
A hypothesis is clean if it names a property that is:
- Positive (asserts something is true, not that something is not the case).
- Defined by an existential ("n = 2k for some integer k" — something exists that we can pluck out).
- Algebraically unpackable (the definition gives you an equation you can manipulate).
Examples of clean hypotheses:
- "n is even" → n = 2k for some integer k. Positive. Existential. Algebraic.
- "a \mid b" → b = aq for some integer q. Positive. Existential. Algebraic.
- "r is rational" → r = p/q for integers p, q with q \neq 0. Positive. Existential. Algebraic.
- "A \subseteq B" → every x \in A is in B. Positive universal; trivial to unpack when you pick a generic x.
- "f is continuous at a" → \varepsilon-\delta statement. Technically positive; unpackable.
When the hypothesis is clean, unpacking the definition yields an equation or a concrete membership statement. From there, direct manipulation usually carries you to the conclusion.
What a "dirty" hypothesis looks like
A hypothesis is dirty (and direct proof tends to stall) if it is:
- Negative — "n is not divisible by 3", "there is no integer k with...". Negatives do not give you a handle to manipulate.
- Disjunctive — "n is 1 or n is prime". You have to case-split before you can do anything.
- Existential in a complicated way — "there exists a function f with property X". Sometimes doable, often requires construction.
When the hypothesis is dirty, direct proof often stalls; you may need contrapositive (which swaps P and \neg Q, possibly making them clean) or contradiction.
The recognition table
| Hypothesis form | Clean? | First try |
|---|---|---|
| "n is even" | Yes | Direct |
| "n is odd" | Yes | Direct |
| "a divides b" | Yes | Direct |
| "r is rational" | Yes | Direct |
| "A \subseteq B" | Yes | Direct |
| "f(x) = g(x) for all x" | Yes (universal-positive) | Direct |
| "n is not even" | Dirty (negation) | Contrapositive or cases |
| "r is irrational" | Dirty (negation of rational) | Contradiction often works |
| "there is no integer k with..." | Dirty (existential negation) | Contradiction |
| "P or Q" | Dirty (disjunction) | Cases |
The left column is what you read in the problem statement. The right column is your first technique to try. If it stalls, switch.
Why 80%
In a typical undergraduate proof-writing course, about 80% of direct-proof-suitable homework problems have one of the clean hypotheses above. This is not a coincidence: clean hypotheses are the ones textbooks reach for when introducing direct proof, and they dominate the early chapters of number theory, algebra, and analysis. The remaining 20% — negations, irrationality, impossibility statements — live in later chapters where contradiction and contrapositive take centre stage.
If you recognise a clean hypothesis, the odds heavily favour direct. Default to it. You can always switch if the bridge from P to Q stalls, but you should not spend energy picking a technique before you have even tried the cheapest one.
Why defaulting is efficient: picking a technique takes cognitive effort. If you spend 10 minutes trying direct and it fails, you have lost 10 minutes. If you spend 10 minutes agonising over which technique to use and then 10 minutes executing, you have lost 20 — and the direct attempt was probably going to work anyway.
A recognition dial
The recognition checklist
When you read "If P then Q," run three checks on P in order. The first "yes" gives you the recommendation.
- Is P a positive, unpackable property? (even, odd, divisible, rational, subset, continuous, differentiable). If yes → try direct.
- Is P a negation of a clean property? (not even, not divisible, irrational). If yes → try contrapositive (it often un-negates things), or contradiction.
- Is P an existential that will be hard to exploit directly? (there exists f such that...). If yes → the proof probably hinges on a specific construction; examine whether direct, contradiction, or a case split is right.
Most school-level problems land on step 1 and stop there.
A walked recognition
Claim: if n is divisible by 6, then n is divisible by 3.
- Hypothesis: "n is divisible by 6." Clean. Unpack to n = 6k.
- First try: direct.
- n = 6k = 3(2k). Since 2k is an integer, 3 \mid n. \blacksquare
Done in one line of algebra. Recognition was instant.
Claim: if r is irrational, then r^2 is irrational.
Wait — is this even true? No: r = \sqrt{2} is irrational, but r^2 = 2 is rational. So the claim is false; you would try to find a counterexample instead. The recognition step ruled out direct proof and ruled out trying to prove the claim at all.
Claim: \sqrt{2} is irrational.
- Hypothesis: none explicit; the conclusion is a negative statement (not rational).
- Recognition: direct proof will stall because you have no clean hypothesis to unpack.
- First try: contradiction. Assume \sqrt{2} is rational and derive an impossibility.
The recognition step saved you from a doomed direct-proof attempt.
What to do when the recognition mispredicts
Sometimes you read a problem, see a clean hypothesis, reach for direct — and the bridge does not appear. That is okay. Direct proof is cheap: two lines in ("Assume P. By definition, ...") and you can tell whether you have a path. If you do not, switch:
- If the conclusion Q is a negation, try contrapositive — the contrapositive \neg Q \Rightarrow \neg P might have cleaner hypotheses than the original.
- If the conclusion is a statement of impossibility or uniqueness, try contradiction.
- If P quantifies over positive integers and there is a recursive structure ("n! grows faster than ..."), try induction.
Switching early and often is a valid proof-writing strategy. The cost of a failed direct attempt is two wasted lines; the cost of never attempting direct is often a much longer, much uglier proof.
The short summary
- Clean hypothesis signals: even, odd, divisible, rational, subset, continuous, differentiable — positive, unpackable, existential.
- First-try rule: for a clean hypothesis, try direct proof. Success rate around 80% at school level.
- Dirty hypothesis signals: not even, not divisible, irrational, there is no integer such that... — start with contrapositive or contradiction.
- Cheap to abandon: if direct stalls after unpacking, switch techniques without guilt.
- Recognition is instant once you practise it on a few examples; make it your reflex.
"Clean hypothesis, try direct first" is one of the highest-value heuristics in proof-writing. It turns the technique-choice question from paralysing into routine, and it lets your energy go where it belongs — into the bridge step, where the real thinking happens.
Related: Mathematical Proof — Direct Proof · Which Proof Technique Flowchart · First Move — Assume P and Unpack · Proof by Contradiction