# Quantifiers - Changing the positions of different types of quantifiers

• Aug 26th 2009, 07:45 AM
kevinlightman
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.
• Aug 26th 2009, 08:04 AM
Plato
Quote:

Originally Posted by kevinlightman
∃ 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 $\displaystyle \left( {\exists x} \right)\left( {\forall y} \right)\left[ {P(x,y)} \right] \Rightarrow \left( {\forall y} \right)\left( {\exists x} \right)\left[ {P(x,y)} \right].$