Practice question · Select all that apply
Select every statement that correctly expresses the negation of 'for all x, P(x)'.
Hints
- The negation flips the quantifier and negates the inside, so it should be an existential claim about a failure.
- Two of the options say 'none', which is a much stronger statement than 'not all'.
Show the answer
- C. it is not the case that every x satisfies P(x)
- E. there exists x with NOT P(x)
Why
'There exists x with not P(x)' is the formal negation and 'it is not the case that every x satisfies P' is the same claim in words. Options 2 and 5 both say no x satisfies P, which is stronger than needed, and option 1 is not a negation at all. 'Not all' means 'at least one is not', never 'none is'.
Practise Predicate Logic and Quantifiers
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