A student tries to prove 'If n is even, then n² is even.' They write: 'Assume n is even. Since even numbers squared are even, n² is even. Therefore if n is even, n² is even.' What is wrong with this proof?
AThe student stated the hypothesis incorrectly — they should have assumed n² is even
BThe student used the conclusion ('even numbers squared are even') as a step inside the proof, which assumes what was supposed to be demonstrated
CThe student needed to prove a lemma first before this argument could work
DThe student forgot to specify that n must be a positive integer
This is a circular argument — the most common beginner proof error. The claim 'even numbers squared are even' is exactly the statement the student is supposed to prove. Using it as a step inside the proof assumes the conclusion, making the argument logically empty. A valid proof must derive the conclusion from the hypothesis using only things already established: assume n is even, write n = 2k, compute n² = 4k² = 2(2k²), conclude n² is even by definition.
Question 2 Multiple Choice
After proving a major theorem about the convergence of infinite series, a mathematician immediately observes that a simpler special case follows almost automatically as a consequence. This simpler result is called:
AA lemma — a helper result proved specifically to support the main theorem
BAn axiom — a foundational assumption the theorem rests on
CA corollary — a result that follows easily as a consequence of a just-proved theorem
DA hypothesis — the conditional assumption from which the theorem was derived
A corollary is a result that follows readily from a theorem that has just been proved — it extends the theorem's reach almost for free. A lemma, by contrast, is proved before the main theorem to support it. An axiom is assumed without proof as a foundational rule. A hypothesis is the 'if' part of a conditional statement. These distinctions reflect the architecture of mathematical knowledge: theorems rest on lemmas, and corollaries extend their reach.
Question 3 True / False
A lemma differs from an axiom in that a lemma must be proved, even though both serve as stepping stones in a larger argument.
TTrue
FFalse
Answer: True
An axiom is assumed to be true without proof — it is a starting point, a foundational rule of the game. A lemma is a proved result: it is a full theorem, just one proved specifically because it is needed to establish something larger. Both play supporting roles in mathematical arguments, but only axioms are accepted without demonstration.
Question 4 True / False
In a direct proof of 'If P, then Q,' the mathematician begins by assuming Q is true and then derives P — this is the standard structure of a direct proof.
TTrue
FFalse
Answer: False
In a direct proof, you assume the hypothesis P and derive the conclusion Q. Assuming Q and deriving P is the structure of a proof by contrapositive (where you prove ¬P → ¬Q instead) or part of proof by contradiction. Reversing hypothesis and conclusion is a common beginner error — often confused with contrapositive reasoning. In a direct proof, the flow is always: start with P, apply logical steps, arrive at Q.
Question 5 Short Answer
Why is writing a mathematical proof described as a 'communication to a reader' rather than just a chain of private reasoning, and what does this mean for how proofs should be structured?
Think about your answer, then reveal below.
Model answer: A proof's purpose is not only to convince yourself but to allow any qualified reader to independently verify every step. This means logical transitions must be made explicit using precise language ('therefore,' 'it follows that,' 'assume,' 'we have shown'), the hypothesis and conclusion must be clearly separated so the reader knows what is being established, and each step must be justified by prior results, definitions, or axioms. Private reasoning can take shortcuts; a written proof cannot, because it must be independently checkable.
This is what distinguishes mathematics from other fields of argument. In mathematics, a claim is not accepted because the arguer is authoritative or the reasoning sounds plausible — it is accepted only when the logical structure is transparent enough that any qualified reader can trace every step. This standard is demanding precisely because it is what makes mathematical truth reliable: the proof stands or falls on its logic alone, independent of who wrote it.