Practice question · Put in order
Order the steps that negate 'for every integer x there exists an integer y with y > x'.
- Negate the outer quantifier: there exists x, NOT (there exists y, y > x)
- Negate the inner quantifier: there exists x, for all y, NOT (y > x)
- Simplify the inner predicate: there exists x, for all y, y is at most x
- Write the statement as: for all x, (there exists y, y > x)
Hints
- Negation moves inward one quantifier at a time, flipping each as it passes.
- Only simplify the predicate once the quantifiers are all dealt with.
Show the answer
- Write the statement as: for all x, (there exists y, y > x)
- Negate the outer quantifier: there exists x, NOT (there exists y, y > x)
- Negate the inner quantifier: there exists x, for all y, NOT (y > x)
- Simplify the inner predicate: there exists x, for all y, y is at most x
Why
Push the negation through one layer at a time: each quantifier it crosses flips, and only at the end does it act on the predicate itself. The result claims a largest integer exists, which is false, correctly making the original statement true.
Practise Quantifiers and Predicates
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