Practice question · Match the pairs
Match each quantified statement to its correct negation.
- for all x, P(x)
- there exists x with P(x)
- every student passed
- some student passed
- there exists x with NOT P(x)
- for all x, NOT P(x)
- no student passed
- at least one student did not pass
Hints
- Negation flips the quantifier and negates what is inside it.
- The negation of a universal claim is an existential one, and vice versa, never the same quantifier twice.
Show the answer
- for all x, P(x) → there exists x with NOT P(x)
- there exists x with P(x) → for all x, NOT P(x)
- every student passed → at least one student did not pass
- some student passed → no student passed
Why
Negating 'all are P' gives 'at least one is not P', which is exactly why one counterexample refutes a universal claim; negating 'some is P' gives 'none is P'. Writing the negation of 'all are P' as 'all are not P' is the standard error, and it is a much stronger claim than the truth requires.
Practise Predicate Logic and Quantifiers
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