Once you have decided that proof by contradiction is the right technique, the very first line of the proof is not a choice — it is a ritual. You write:
"Suppose, for the sake of contradiction, that ..."
…and then you carefully write down the negation of the statement you are trying to prove. That is it. No creativity, no variation. Every correct contradiction proof begins this way.
The catch is in the word carefully. The second half of the sentence — the negation — is where sloppy proofs go wrong before they even get off the ground. Negating a mathematical statement is mechanical, but the mechanics trip up most students on their first few attempts. This article drills the two halves of the opening line: the ritual on the left, and the exact-negation rules on the right.
The ritual phrase
Write the opening exactly like this, or a close equivalent:
- "Suppose, for the sake of contradiction, that …"
- "Assume, for contradiction, that …"
- "Towards a contradiction, suppose …"
All three are interchangeable. The purpose of the phrase is threefold:
- Announce the technique. The reader knows immediately that the proof is contradiction-based and will end in an impossibility.
- Mark the hypothesis as tentative. Everything inside the "suppose" block is not a fact — it is a candidate that will eventually be demolished. Subsequent manipulations must respect that this is a hypothetical world.
- Force you to write the negation explicitly. The ritual phrase demands a grammatical object — you cannot skip writing what you are negating.
Why the ritual matters: a proof that drifts into contradiction-style reasoning without announcing it is confusing to read. The reader has to hunt for where the contradiction technique began and where the "real" world resumes. The opening phrase is a small investment in clarity that pays off every time someone (including future-you) reads the proof.
Negating the goal — the mechanical rules
The harder half of the opening line is writing down the actual negation. Here are the rules you apply without thinking, every time.
| If the goal says... | The negation is... |
|---|---|
| P (a simple statement) | \lnot P ("P is not true") |
| "for all x, Q(x)" | "there exists x with \lnot Q(x)" |
| "there exists x with Q(x)" | "for all x, \lnot Q(x)" |
| "if P then Q" | "P is true and Q is false" |
| A \land B | \lnot A \lor \lnot B |
| A \lor B | \lnot A \land \lnot B |
| "x is irrational" | "x = p/q for some integers p, q with q \neq 0" |
| "no x satisfies P" | "some x satisfies P" |
The two most error-prone rules are the quantifier flip (row 2 and row 3) and the implication rule (row 4). Students routinely negate "if P then Q" as "if P then \lnot Q" — which is wrong. The correct negation is "P and \lnot Q" — both halves true simultaneously.
Why the implication negation is what it is: an implication "if P then Q" fails in exactly one situation: P is true and Q is false. (If P is false, the implication is vacuously true; if Q is true, the implication holds regardless of P.) So the only way to deny the implication is to assert P \land \lnot Q. This is a rule worth memorising cold — every contradiction proof of an implication leans on it.
Three goal types, three openings
Let the goal have one of three common forms. Here is the opening line you write for each.
Goal Type 1: existence denial. "Prove that there is no integer n with n^2 = 2."
Opening:
"Suppose, for contradiction, that there exists an integer n with n^2 = 2."
The negation flipped no (universal negation) into there exists.
Goal Type 2: universal property. "Prove that for every integer n, n^2 \geq 0."
Opening:
"Suppose, for contradiction, that there exists an integer n with n^2 < 0."
The negation flipped for every into there exists and flipped the inequality.
Goal Type 3: implication. "Prove that if n^2 is even, then n is even."
Opening:
"Suppose, for contradiction, that n^2 is even and n is odd."
The negation of the implication kept the hypothesis and negated the conclusion, joining them with and. This is the structure that "if P then Q" demands.
Negation dial
Drag the dot to cycle through the three goal types. The readout shows the goal on top and the opening line of the proof on the bottom, after applying the negation rule.
The two most common opening mistakes
Mistake 1: Forgetting to negate the quantifier. A student writes "Suppose, for contradiction, that for every integer n, n^2 < 0." This is not the negation of "for every integer n, n^2 \geq 0." — it is a stronger universal claim. The correct negation is "there exists an n with n^2 < 0." Always flip \forall to \exists (and vice versa) and move the negation inside.
Mistake 2: Negating the implication wrong. The goal is "if P then Q." The student writes "Suppose P is false and Q is false," or "Suppose P is true and Q is true," or "Suppose \lnot P implies \lnot Q." All wrong. The correct negation is P \land \lnot Q — the hypothesis is true and the conclusion is false. This single rule is responsible for more botched contradiction proofs than any other.
Practising the ritual
Goal: "Prove that \sqrt{3} is irrational."
Opening (unpacking irrational as not rational):
"Suppose, for contradiction, that \sqrt{3} is rational. Then there exist integers p, q with q \neq 0 and \sqrt{3} = p/q."
The ritual phrase appears, the negation is explicit, and the existence of p, q is now something you can manipulate algebraically.
Goal: "Prove that if n is prime and n > 2, then n is odd."
Opening (negating the implication):
"Suppose, for contradiction, that n is prime, n > 2, and n is even."
The hypothesis (n prime and n > 2) is preserved; only the conclusion (n is odd) is negated (to n is even). The proof continues by noting that an even number > 2 is divisible by 2, contradicting primality.
Goal: "Prove that for every real x, if x > 0 then x + 1/x \geq 2."
Opening (negating universal + implication):
"Suppose, for contradiction, that there exists a real x > 0 with x + 1/x < 2."
The universal became existential; the implication's hypothesis was kept; the conclusion was negated (inequality flipped). Three moves, all mechanical.
A useful sanity check
After writing the opening line, read it aloud and ask: "Is this the opposite of what I was asked to prove?" If the answer is yes, you are set up correctly. If the answer is "sort of" or "I think so," the negation is probably wrong. Common symptoms of a bad negation:
- You negated twice, accidentally restating the original goal.
- You negated the hypothesis instead of the conclusion in an implication.
- You kept the quantifier unchanged instead of flipping it.
When in doubt, write out the logical symbols: \lnot (\forall x \, P(x)) becomes \exists x \, \lnot P(x), mechanically. Then translate back to English.
Why this sanity check is worth 30 seconds: a proof built on a wrong negation cannot succeed. You will either derive nothing (dead end) or derive something, but the "something" you derive will not imply the original goal. Catching the negation error at line 1 saves half a page of wasted algebra.
The ritual in short
- Write "Suppose, for contradiction, that ..." — no variations needed.
- After it, write the negation of the goal, using the mechanical rules (quantifier flip; implication \to P \land \lnot Q).
- Read aloud to confirm you have stated the opposite.
- Then — and only then — start deriving consequences.
The opening line is the most mechanical part of a contradiction proof. That is a feature, not a limitation: once the ritual is automatic, your creativity is freed up for the interesting middle where the contradiction actually appears.
Related: Proof by Contradiction · Negative Keywords Signal Contradiction · Quantifier Flipper — ∀ ⇔ ∃ with Negation · Assume the Negation — Preserve Meaning