The negation of "for all x there exists y with P(x,y)" is "there exists x such that for all y, not P(x,y)". Why do the quantifiers flip as well as the predicate?

Question

Accepted Answer

Because denying 'every' needs one failure and denying 'some' needs all

Answer

Because only the innermost quantifier changes and the outer one is unaffected

Answer

Because negation reverses the order in which the variables were introduced

Answer

Because the predicate’s negation makes the quantifiers redundant

The negation of "for all $x$ there exists $y$ with $P(x,y)$ " is "there exists $x$ such that for all $y$ , not $P(x,y)$ ". Why do the quantifiers flip as well as the predicate?

Practise Quantifiers and Predicates

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The negation of "for all xxx there exists yyy with P(x,y)P(x,y)P(x,y)" is "there exists xxx such that for all yyy, not P(x,y)P(x,y)P(x,y)". Why do the quantifiers flip as well as the predicate?

Practise Quantifiers and Predicates

More questions on Quantifiers and Predicates

The negation of "for all $x$ there exists $y$ with $P(x,y)$ " is "there exists $x$ such that for all $y$ , not $P(x,y)$ ". Why do the quantifiers flip as well as the predicate?