Which of the following correctly states the negation of 'Every student in the class passed the exam'?
AEvery student in the class failed the exam
BNo student in the class passed the exam
CAt least one student in the class did not pass the exam
DMost students in the class did not pass the exam
The original statement is ∀x P(x). Its negation is ∃x ¬P(x) — 'there exists at least one student who did not pass.' Option A ('every student failed') is ∀x ¬P(x), which is far stronger than the negation requires: it says everyone fails, when even a single failure suffices to make the original claim false. Option B is equivalent to option A. Negation flips the quantifier from ∀ to ∃ AND negates the predicate — it does not simply negate the predicate while leaving the quantifier unchanged.
Question 2 Multiple Choice
What is the correct negation of the statement ∀x ∃y (x + y = 0)?
A∃x ∀y (x + y = 0)
B∀x ∃y ¬(x + y = 0)
C∃x ∀y (x + y ≠ 0)
D∀x ∀y (x + y ≠ 0)
Apply the negation rule from outside in: ¬∀x becomes ∃x, then ¬∃y becomes ∀y, and finally the predicate is negated: (x + y = 0) becomes (x + y ≠ 0). Result: ∃x ∀y (x + y ≠ 0). Option A flips only the first quantifier but leaves ∃y unchanged. Option B negates only the predicate without flipping ∃y. Option D negates the predicate but flips neither quantifier. Each quantifier must be flipped in turn as the negation pushes inward.
Question 3 True / False
The negation of 'There exists a prime number greater than 10' is 'All prime numbers are at most 10.'
TTrue
FFalse
Answer: True
The original statement is ∃x P(x). Its negation is ∀x ¬P(x). 'There exists a prime > 10' negates to 'for all numbers that are prime, they are ≤ 10' — i.e., all prime numbers are at most 10. The quantifier correctly flips from ∃ to ∀ and the predicate is negated.
Question 4 True / False
The negation of 'Most mathematicians are brilliant' is 'No mathematicians are brilliant.'
TTrue
FFalse
Answer: False
¬(∀x P(x)) ≡ ∃x ¬P(x) — the negation is 'there exists at least one mathematician who is not brilliant.' 'No mathematicians are brilliant' is ∀x ¬P(x), which makes the much stronger claim that every single mathematician fails the property. This is the most common error: applying ¬ only to the predicate while leaving the ∀ unchanged. Negation must flip the quantifier from ∀ to ∃.
Question 5 Short Answer
Explain why the negation of 'All S are P' is not 'All S are not-P', and state the correct negation.
Think about your answer, then reveal below.
Model answer: The negation of a universal claim requires only one counterexample. 'All S are P' (∀x P(x)) is false whenever even a single S fails to be P. So its negation is 'There exists at least one S that is not P' (∃x ¬P(x)). 'All S are not-P' (∀x ¬P(x)) asserts that every element fails — a much stronger claim that is neither required nor implied by the original being false. Negation flips the quantifier (∀ to ∃) AND negates the predicate; it does not simply negate the predicate.
A useful test: if even one S is P, the original claim is false, but 'all S are not-P' is also false. So 'all S are not-P' cannot be the negation of 'all S are P.' A statement and its negation must have opposite truth values in every case.