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Math Help - Quantifiers - Changing the positions of different types of quantifiers

  1. #1
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    Quantifiers - Changing the positions of different types of quantifiers

    I am trying to find two predicates that don't work when the positions of different types of quantifiers are switched.
    The first one is when:

    ∀x ∃ y P( x, y ) is true but ∃ y ∀ x P( x, y ) is false

    So I though when you let P(x, y) denote x < y would satisfy that one. The second predicate I need is when:

    ∃ x ∀ y P( x, y ) is true but ∀y ∃ x P( x, y ) is false.

    A simple answer to this would be preferable like my previous one.
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  2. #2
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    Quote Originally Posted by kevinlightman View Post
    ∃ x ∀ y P( x, y ) is true but ∀y ∃ x P( x, y ) is false.
    A simple answer to this would be preferable like my previous one.
    You will never find an example because \left( {\exists x} \right)\left( {\forall y} \right)\left[ {P(x,y)} \right] \Rightarrow \left( {\forall y} \right)\left( {\exists x} \right)\left[ {P(x,y)} \right].
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