You want to prove 'if A then B.' In setting up a proof by contradiction, what should you assume?
AAssume ¬A (the hypothesis is false)
BAssume ¬B (the conclusion is false)
CAssume A ∧ ¬B (the hypothesis is true and the conclusion is false)
DAssume ¬A ∧ ¬B (both hypothesis and conclusion are false)
To prove 'if A then B' by contradiction, you assume the negation of the entire conditional. The negation of 'if A then B' is 'A and not-B' — not simply ¬B. Assuming only ¬B is a common error that proves something weaker or creates a circular argument. The proof then derives a contradiction from A ∧ ¬B, establishing that this combination is impossible, and therefore 'if A then B' must hold.
Question 2 Multiple Choice
In the classic proof that √2 is irrational, the contradiction reached is:
A√2 turns out to equal a specific rational number, contradicting the assumption it was irrational
BThe denominator q turns out to equal zero, making the fraction undefined
CBoth p and q must be even, contradicting the assumption that p/q was in lowest terms
Dp² = 2q² has no integer solutions, directly contradicting the assumption
The proof assumes √2 = p/q in lowest terms (no common factors), then shows p must be even (so p = 2k), substitutes to get q² = 2k², which forces q to also be even. But if both p and q are even, they share the factor 2 — directly contradicting the assumption that p/q was already in lowest terms. This contradiction discharges the assumption, establishing that √2 cannot be rational.
Question 3 True / False
In a proof by contradiction, the contradiction derived should take the explicit form of a statement P being asserted both true and false simultaneously (P ∧ ¬P).
TTrue
FFalse
Answer: False
The contradiction can be any statement already known to be false — not just an explicit P ∧ ¬P. It might be a violation of a previously proven theorem, a consequence like 0 = 1, or a derived result that contradicts a known fact (like 'both p and q are even' contradicting 'p/q is in lowest terms'). The only requirement is that the derived statement is demonstrably impossible, given what was already established before the proof began.
Question 4 True / False
A proof by contradiction can establish that an object exists without constructing or exhibiting the object explicitly.
TTrue
FFalse
Answer: True
This is what makes contradiction proofs non-constructive. The proof of √2's irrationality, for example, proves a negative — that no rational number equals √2 — without building anything. Existential claims can also be proven this way: assume the object does not exist, derive a contradiction, conclude it must exist. The proof demonstrates truth by eliminating the alternative, not by exhibiting a positive construction. This is what distinguishes contradiction proofs from direct proofs.
Question 5 Short Answer
Why must you negate the *entire* goal when setting up a proof by contradiction, and what goes wrong if you only negate part of it?
Think about your answer, then reveal below.
Model answer: You must negate the entire goal because the proof works by showing the negation leads to a contradiction — if you only negate part of the goal, you are assuming something different from 'the goal is false,' so any contradiction you derive doesn't establish the full original statement. For a conditional 'if A then B,' negating only B assumes 'not-B' without assuming A, which means contradictions you reach might depend on the absence of A rather than on A ∧ ¬B — proving a weaker or different result.
Logical hygiene here is essential. The proof's validity depends on the chain: (negation of goal) → contradiction → goal is true. If the negation is incomplete or incorrect, the chain breaks. Partial negations lead to proofs that feel valid but actually establish something other than the intended claim — a subtle error that can be difficult to spot after the fact.