Practice question · True or false
If 'there exists x with P(x)' is true, then 'for all x, P(x)' must also be true.
Hints
- Try a predicate that holds for one value in the domain but not another.
- One witness says nothing about the rest of the domain.
Show the answer
False
Why
False. Over the integers, 'there exists x with x = 0' is true while 'for all x, x = 0' is plainly false. An existential claim needs one witness; a universal claim needs the entire domain. The implication runs the other way, over a non-empty domain, universal implies existential.
Practise Quantifiers and Predicates
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