Most proof-technique questions have no clean one-line answer — you read the claim, try direct, switch if stalled. But there are four problem types where the decision is close to automatic. When you see one of these, don't deliberate. Proof by contradiction is the default first attempt, and it almost always works.
The four types are:
- Irrationality — "prove \alpha is irrational" (or transcendental, or not expressible as...)
- Non-existence — "no integer n satisfies..."
- Uniqueness — "there is exactly one x with property P"
- Infinitely many — "there are infinitely many primes", "infinitely many n satisfy..."
This article walks through why each one snaps so naturally into the contradiction template, so you can pattern-match them in milliseconds.
Type 1: Irrationality
Signal: the conclusion is "\alpha is irrational" or "\alpha cannot be written as p/q."
Why contradiction: Irrational is defined as not rational. A negative definition does not hand you anything to manipulate directly. Contradiction flips it: assume \alpha = p/q with p, q integers (usually in lowest terms), and then the algebra has a concrete equation to chew on.
Opening: "Suppose, for contradiction, that \alpha = p/q for some integers p, q with q \neq 0." (Optionally add lowest terms — see why lowest terms matters.)
Typical engine: square both sides, derive divisibility, find a contradiction (either both p, q divisible by the same prime — contradicting lowest terms — or an infinite descent).
Examples covered by this template: \sqrt{2}, \sqrt{3}, \sqrt{n} for any non-perfect-square integer n, \log_2 3, e, \pi (the last two are harder but structurally identical).
Type 2: Non-existence
Signal: "There is no x such that...", "P(x) has no solutions", "no integer of the form...", "f is not..."
Why contradiction: you cannot check every possible x directly — there are usually infinitely many candidates. Contradiction picks just one (the hypothetical x that does satisfy the forbidden property) and shows it cannot actually exist.
Opening: "Suppose, for contradiction, that there exists an x with..."
Typical engine: use the properties of x (from the assumed condition) together with the properties of its domain to derive something impossible. Often the contradiction is with the definition of the domain (e.g., x would need to be both positive and negative, or both integer and non-integer).
Examples: "There is no integer n with n^2 \equiv 2 \pmod 4.", "No polynomial of degree n \geq 1 has more than n roots.", "No continuous function on [0, 1] is unbounded."
Type 3: Uniqueness
Signal: "There is exactly one x with property P", "x is the unique...".
Why contradiction: uniqueness has two parts — existence and uniqueness. Existence is usually handled directly (you produce x). The uniqueness part is where contradiction shines: assume there are two distinct objects x_1, x_2 both with property P, and show they must be equal. The contradiction is then with the assumption of distinctness.
Opening (for the uniqueness half): "Suppose, for contradiction, that x_1 \neq x_2 are two distinct elements with property P."
Typical engine: use P(x_1) and P(x_2) together with some algebraic operation — often x_1 - x_2 or x_1 / x_2 — to derive x_1 = x_2, contradicting the distinctness assumption.
Examples: "Given two points in the plane, there is exactly one line through them.", "The equation a x = b with a \neq 0 has a unique solution.", "Every non-zero integer has a unique prime factorisation (up to order)."
Type 4: Infinitely many
Signal: "There are infinitely many primes", "infinitely many n satisfy...", "the sequence (a_n) has infinitely many..."
Why contradiction: assuming the conclusion is false gives you a finite set to work with — "suppose only finitely many satisfy..." — and finite sets are easier to manipulate algebraically. You can list them, multiply them, add 1, whatever the algebra wants. The contradiction is usually that the finite list is forced to produce one more element not on it.
Opening: "Suppose, for contradiction, that only finitely many ... satisfy... List them: x_1, x_2, \ldots, x_n."
Typical engine: construct a new object from the list that must also satisfy the property but cannot be on the list. The construction is the creative step; the contradiction is mechanical.
Examples: "Infinitely many primes" (Euclid: form p_1 p_2 \cdots p_n + 1); "Infinitely many primes of the form 4k + 3" (similar construction); "Infinitely many n with \sqrt{n} irrational" (almost all n).
The recognition grid
| Claim type | Keyword | Opening | Engine |
|---|---|---|---|
| Irrationality | irrational, cannot be p/q | Suppose \alpha = p/q | Divisibility, prime propagation |
| Non-existence | no, does not exist | Suppose x exists with P | Properties of x vs. domain |
| Uniqueness | exactly one, the unique | Suppose x_1 \neq x_2 both satisfy P | Derive x_1 = x_2 |
| Infinitely many | infinitely many | Suppose finitely many: x_1, \ldots, x_n | Build a new x not on the list |
Four rows, four patterns. Memorise the grid, and the technique selection for a huge fraction of problems becomes instant.
The classifier
Drag the dot to cycle through four sample claims. The readout identifies the type, names the opening, and pairs it with the engine.
Why the default works so often
All four types share a common structure: the conclusion is indirect — it says something does not exist, or is unique, or is not rational, or cannot be listed finitely. Direct proofs are awkward for indirect conclusions because they demand you produce something (a construction, a universal witness), and what you need to produce is literally the absence of something. Contradiction fixes this asymmetry by letting you assume the presence of the forbidden object and reasoning about its properties — turning an absence-proof into a presence-refutation.
Why this is so reliable: the four types are exactly the ones where direct proof has no natural handle. Mathematicians identified this gap centuries ago, and the contradiction template was developed specifically to fill it. That is why the template feels "automatic" — it is, because the problem types that trigger it are ones that resist direct attack by definition.
When the default fails
Even for these four types, contradiction is not always the only path. Two honourable exceptions:
-
Infinitely many with a constructive option. "There are infinitely many even numbers." Trivially direct: exhibit 2, 4, 6, \ldots. No contradiction needed. Whenever a clean constructive sequence is visible, prefer it.
-
Uniqueness via direct algebraic reasoning. "The equation 5x = 10 has a unique solution." Divide both sides by 5 to get x = 2. Uniqueness is visible from the division step. No need to assume two solutions and force them equal.
The template is a default, not a mandate. If direct is short and clean, use it. But if you read an irrationality claim and your first instinct is "let me try to construct...", stop — you are walking away from the template for no reason.
Four claims, four contradictions in four opening lines
Claim 1 (irrationality): "\sqrt{5} is irrational."
Opening: "Suppose \sqrt{5} = p/q in lowest terms. Then 5q^2 = p^2, so 5 \mid p^2, so 5 \mid p: write p = 5k. Then 5q^2 = 25k^2, so q^2 = 5k^2, forcing 5 \mid q. But then 5 divides both p and q — contradicting lowest terms." \blacksquare
Claim 2 (non-existence): "There is no integer n with n^2 \equiv 2 \pmod 4."
Opening: "Suppose such an n exists. If n is even, n = 2m, so n^2 = 4m^2 \equiv 0 \pmod 4. If n is odd, n = 2m+1, so n^2 = 4m^2 + 4m + 1 \equiv 1 \pmod 4. Neither case gives 2. Contradiction." \blacksquare
Claim 3 (uniqueness): "If a \neq 0, the equation ax = b has a unique solution."
Opening (uniqueness part): "Suppose x_1, x_2 both satisfy a x_1 = b and a x_2 = b. Then a x_1 = a x_2, so a(x_1 - x_2) = 0. Since a \neq 0, x_1 - x_2 = 0, i.e., x_1 = x_2 — contradicting x_1 \neq x_2." \blacksquare
Claim 4 (infinitely many): "There are infinitely many primes." (Euclid)
Opening: "Suppose only p_1, \ldots, p_n are prime. Form N = p_1 p_2 \cdots p_n + 1. N has a prime divisor q. But q cannot be any of p_1, \ldots, p_n (because N \equiv 1 \pmod{p_i} for each i). So q is a prime not on the list — contradiction." \blacksquare
Four openings, four contradictions, four finished proofs. Each falls into the template immediately upon reading the claim.
The reflex in short
When you read a problem, scan for one of the four signal types before you start writing anything:
- irrational / cannot be p/q → contradiction, opening: assume rational.
- no / does not exist → contradiction, opening: assume existence.
- unique / exactly one → contradiction for the uniqueness half, opening: assume two distinct.
- infinitely many → contradiction, opening: assume finitely many.
If none of the four applies, try direct first. If any applies, jump to the contradiction template and skip the deliberation. Over a proof-writing career, this one recognition habit will save hundreds of hours of technique-hunting and let you spend that time on the actual mathematics.
Related: Proof by Contradiction · Negative Keywords Signal Contradiction · First Move — Suppose and Negate · Universal Statements — Assume a Counterexample