A mathematical proof is a logical argument that establishes a claim beyond doubt. Most students know this definition. Fewer know what distinguishes a proof that earns full marks from one that earns half or zero, even when the key idea is correct. The failure modes are consistent across courses and across levels.
Failure mode 1 — Starting from what you want to prove
This is the most common error. The student wants to show that A = B, and writes:
A = B (some manipulation) C = C ✓
This is not a proof. It is circular reasoning: you assumed A = B was true in order to show that A = B is true. The conclusion C = C is valid, but it does not tell you anything about whether A = B was true to begin with.
The fix: start from what you know. If you want to show A = B, start from the left side (A), apply legitimate operations, and arrive at B. Or start from a known truth and derive A = B as a consequence.
This requires sometimes working the proof "backwards" in scratch work — assuming the conclusion and figuring out what you need to establish — and then writing the proof "forwards" (starting from known facts).
Failure mode 2 — Treating examples as proofs
"Show that n² + n is always even."
Student writes: "When n = 2, n² + n = 6, which is even. When n = 5, n² + n = 30, which is even. Therefore n² + n is always even."
This earns zero marks for the proof component. Two examples do not prove an assertion about all integers. Even one thousand examples do not. A proof must work for every case simultaneously.
The fix: factor. n² + n = n(n+1). One of any two consecutive integers is even, so n(n+1) is always even (the product of an even number and anything is even). This covers every integer at once.
Failure mode 3 — Not naming the proof method
When you use proof by contradiction, say so: "We proceed by contradiction. Assume..." When you use mathematical induction, state the base case and inductive step explicitly. Examiners look for these structural markers.
A proof that uses the correct ideas but does not name its method often loses one or two marks — sometimes more if the method is an induction and the student has only shown the base case and the inductive step without clearly connecting them.
The fix: before writing, decide which method you are using. Direct proof, proof by contradiction, proof by contrapositive, proof by induction, proof by exhaustion (for finite cases). Write the method name as the first sentence.
Proof by contrapositive
If you want to prove "P implies Q," it is logically equivalent to prove "not Q implies not P." Sometimes the contrapositive is much easier to establish.
Example: "If n² is even, then n is even." Direct proof is awkward because we cannot easily characterise all even perfect squares. Contrapositive: "If n is odd, then n² is odd." Let n = 2k+1. Then n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. Done.
Proof by induction: the two-part obligation
A proof by induction on n ≥ 1 requires two things:
1. Base case: verify the claim for n = 1 (or the smallest relevant n). 2. Inductive step: assume the claim holds for n = k (the inductive hypothesis), and show it holds for n = k+1.
Both are mandatory. A proof that shows only the inductive step — often because the student found it harder to verify the base case — is incomplete and earns partial credit at best.
Frequently Asked Questions
Can I use examples to support a proof?
Examples can motivate or illustrate a proof, but they cannot substitute for it. If your proof includes examples, they should clarify the logic, not replace it. An examiner reading your examples as the proof will award zero for the proof component.
What is the difference between proof by contradiction and proof by contrapositive?
Proof by contrapositive works directly with the contrapositive statement (not Q implies not P). Proof by contradiction assumes both P and not Q simultaneously, and derives an absurdity. Both are valid; the choice depends on which is easier for the specific claim.