3. Give an example of a predicate P(x,y) such that ∃x∀yP(x,y) and ∀y∃xP (x, y) have different truth values.
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Let us give an example of a predicate such that and have different truth values.
Let means and the domains for and are the set of all positive integers.
Since the set does not contain the greatest element, the value of is false. On the other hand, for any let . Then and hence the value of is true.