Four proof techniques handle the overwhelming majority of proofs you will write in school, JEE, and undergraduate mathematics: direct, contrapositive, contradiction, and induction. Each one has a signature — a signal in the problem statement that says "try me first." The flowchart below arranges those signatures so that, for any given claim, you can walk from the stem to the right technique in under ten seconds.
The top branch is always direct proof. Not because it is always easiest, but because when direct works, it is almost always the clearest. Every other technique exists to handle cases where direct is blocked.
The flowchart
The four triggers, one-line each
- Direct. Default. Try it first. Works when the hypothesis unpacks into a concrete form (via definitions) and the conclusion is reachable by algebraic manipulation.
- Contrapositive. Works well when "not Q" gives you a more concrete handle than P. Classic example: "if n^2 is even, then n is even." Directly, you start with "n^2 = 2k" and need to conclude "n is even" — but factoring n^2 = 2k to recover n is hard. Contrapositive: "if n is odd, then n^2 is odd." Now you start with n = 2m + 1 and expand — much easier.
- Contradiction. Shines on non-existence and irrationality claims. "\sqrt{2} is irrational" — impossible to prove by constructing irrationality directly; straightforward by assuming \sqrt{2} = p/q and deriving a contradiction.
- Induction. The only technique that handles "for all n \in \mathbb{N}" claims one n at a time. The claim "1 + 2 + \cdots + n = n(n+1)/2 for all n \ge 1" does not unpack directly — you induct on n.
Walked example: choosing the right technique
Stem. "Prove that if n^2 + 5 is odd, then n is even."
Walk the flowchart.
- Try direct. Assume n^2 + 5 is odd. Then n^2 + 5 = 2k + 1, so n^2 = 2k - 4 = 2(k - 2), hence n^2 is even. You now need: "n^2 even \Rightarrow n even." This is itself a classic problem — and it is most naturally proved by contrapositive. You have unwittingly reduced the problem to a second, trickier one.
- Try contrapositive instead. The contrapositive of the original is: "if n is odd, then n^2 + 5 is even." Let n = 2m + 1. Then n^2 + 5 = (2m+1)^2 + 5 = 4m^2 + 4m + 6 = 2(2m^2 + 2m + 3). So n^2 + 5 is even. Done. \blacksquare
The contrapositive path was shorter and cleaner. Direct was not wrong — it just led to a sub-problem. The flowchart catches this: when the conclusion "n is even" involves a structural fact (even/odd) rather than a direct algebraic consequence, contrapositive is usually easier.
Why direct is the default
Three reasons.
- Clearest prose. A direct proof reads like a story: "we are given P; here is what we can derive; we arrive at Q." No inverted assumptions, no hypothetical impossibilities.
- Easiest to check. Each step follows forward from the previous one. A reader (or you, rereading tomorrow) can verify each line in isolation.
- Most widely applicable. For equalities, for divisibility, for algebraic identities, for geometric statements — direct is the first tool.
Why direct still fails sometimes: when the hypothesis is a vague existential claim ("P is irrational" — no algebraic handle), or when the conclusion demands negative information ("X does not exist"), direct has no starting point to unpack. The other three techniques exist for precisely these cases.
A second walked example
Stem. "Prove that there is no smallest positive rational number."
Walk the flowchart.
- Try direct. You would have to construct an argument that "there is no X with property Y." But there is no hypothesis to unpack — you are asked to prove non-existence. Direct is blocked.
- Try contradiction. Assume for contradiction there is a smallest positive rational — call it r. Then r/2 is also a positive rational (half of a rational is a rational, and half of a positive is positive) and is strictly smaller than r. This contradicts the assumption that r was the smallest. Therefore, no smallest positive rational exists. \blacksquare
The non-existence trigger ("no X with property Y") fires contradiction on the flowchart. Direct had no entry point.
A third walked example — induction
Stem. "Prove that 1 + 2 + \cdots + n = n(n+1)/2 for all n \ge 1."
The "for all n \ge 1" clause is the induction trigger. Direct proof cannot handle a universally quantified statement over the integers in one shot — you would need an argument that works for n = 1, 2, 3, \ldots all at once. Induction gives you a template: prove the base case n = 1, then prove that if the formula holds for some n = k, it holds for n = k + 1. The domino chain does the rest.
The exam reflex
See the stem. Walk the flowchart.
- Direct first. Always. Until you are stuck.
- Stuck on an implication? Try contrapositive — especially if "not Q" is concrete and "not P" is easy to verify.
- Stuck on a non-existence, infinitude, or irrationality claim? Switch to contradiction.
- Stuck on a "for all n" claim? Induction.
Rare cases require more exotic techniques (extremal principle, pigeonhole, double counting), but for school and JEE, the four-branch flowchart covers nearly everything.
Related: Mathematical Proof — Direct Proof · Proof by Contrapositive · Proof by Contradiction · Mathematical Induction