You want to prove: 'If n is odd, then n² is odd.' A student writes: 'Assume n² is odd. Then n² = 2k+1 for some integer k...' and eventually concludes that n must be odd. What error has this student made?
AThey proved the converse (n² odd → n odd) instead of the original statement (n odd → n² odd)
BThey used algebraic manipulation incorrectly
CThey forgot to state that k is an integer
DNo error — they proved the original statement correctly
The student assumed n² is odd (the conclusion) and derived that n is odd (the hypothesis). This proves the converse, q → p, not the original implication p → q. These are different claims — the converse of a true statement is not automatically true. A direct proof of 'if n is odd, then n² is odd' must start by assuming n is odd and then deriving that n² is odd.
Question 2 Multiple Choice
A student proves 'if n is even, then n² is divisible by 4.' Midway through, they write: 'Since n² is divisible by 4, we can write n² = 4m for some integer m, and therefore...' What is the fundamental flaw?
AWriting n² = 4m is not a valid algebraic substitution
BThe theorem is false for some values of n
CThe student is assuming the conclusion before proving it, making the argument circular
DThey should have used proof by contradiction instead
A direct proof must derive the conclusion; it cannot assume it. Writing 'since n² is divisible by 4' as a step inside the proof that n² is divisible by 4 is circular reasoning — the thing being proved is smuggled in as a premise. The only thing you are allowed to assume at the start is the hypothesis (n is even). Every subsequent step must follow from previous ones.
Question 3 True / False
In a valid direct proof of 'if P, then Q,' the only statement you are permitted to assume at the very start is P.
TTrue
FFalse
Answer: True
This is the defining constraint of a direct proof. You assume the hypothesis P and nothing else. All other statements — intermediate conclusions, substitutions, applications of definitions — must be derived from P using logic, definitions, and previously established theorems. Assuming anything beyond P (especially Q itself) invalidates the proof.
Question 4 True / False
Successfully proving 'if Q, then P' also proves 'if P, then Q.'
TTrue
FFalse
Answer: False
Proving 'if Q, then P' proves the converse of 'if P, then Q' — a completely different statement. An implication and its converse are logically independent: one can be true while the other is false. For example, 'if n is even, then n² is even' is true, but its converse 'if n² is even, then n is even' requires a separate proof. Proving the converse is one of the most common errors in beginning proof-writing.
Question 5 Short Answer
Why does assuming the conclusion at the start of a direct proof invalidate the argument, even if every subsequent step is logically valid?
Think about your answer, then reveal below.
Model answer: A proof is meant to establish that Q must follow from P alone. If you assume Q as a premise, you have added it to your starting assumptions — so your argument only shows that Q follows from {P, Q}, which is trivially true and tells you nothing. The whole point of the proof is to show Q is necessary given P. Assuming Q makes the argument circular: the conclusion depends on itself.
This is the distinction between a proof and a verification. A verification starts with Q and checks consistency; a proof starts only with P and derives Q. Any step that introduces Q as an assumption — even implicitly — collapses the proof into circular reasoning. The practical warning sign: if you find yourself writing the conclusion in the middle of the argument, check whether you are using it as a premise.