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I must be missing something here because these two statements look identical to me in regards to their truth tables. To me, ∃x∀y P(x , y) and ∀y∃x P(x , y) are logically equivalent...

a) What is the difference between the quantification ∃x∀y P (x , y) and ∀y∃x P (x , y), where P (x , y) is a predicate?

b) Give an example of a predicate P (x , y) such that ∃x∀y P (x , y) and ∀y∃x P (x , y) have different truth values.

  • Here's a start: http://math.stackexchange.com/questions/201051/is-the-order-of-universal-existential-quantifiers-important; and also http://math.stackexchange.com/questions/225654/are-these-two-predicate-statements-equivalent-or-not – Asaf Karagila Apr 10 '13 at 18:06
  • Thanks, @Asaf ! I know they're out there ;-) – amWhy Apr 10 '13 at 18:08
  • I encourage the next votes to close to use other links than the two already mentioned in the dialog. – Asaf Karagila Apr 10 '13 at 18:08

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a)"There exists x such that for all y ..." is much stronger than the second one.In other

words, this implies the other since in the second statement you may choose different x values

for different y values.

b)There exists x such that for all y :$x>y$ $\hspace{17mm} $ (F)

For every y there is x such that :$x>y$ $\hspace{22mm} $ (T)

Halil Duru
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