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Answer :
∃x ∀y P(x, y) is equivalent to ∀y ∃x P(x, y), where P(x, y) is a predicate involving variables x and y.
To demonstrate that the two statements are equivalent, let's break down their meanings:
∃x ∀y P(x, y) means "There exists an x such that for all y, P(x, y) is true."
∀y ∃x P(x, y) means "For all y, there exists an x such that P(x, y) is true."
To show their equivalence, we need to prove that if one statement is true, the other is also true, and vice versa.
Assume ∃x ∀y P(x, y) is true. This means that there exists at least one value of x such that for all possible values of y, P(x, y) is true. Now, let's consider the statement ∀y ∃x P(x, y).
Since the quantifiers are swapped, it states that for all possible values of y, there exists at least one value of x such that P(x, y) is true.
This is essentially the same as the original statement, where we have one x value that satisfies the predicate for all y values. Therefore, if ∃x ∀y P(x, y) is true, then ∀y ∃x P(x, y) is also true.
Conversely, assume ∀y ∃x P(x, y) is true. This means that for all possible values of y, there exists at least one value of x such that P(x, y) is true. Now, let's consider the statement ∃x ∀y P(x, y).
This statement states that there exists at least one value of x such that for all possible values of y, P(x, y) is true. Since we already know that for all y, there exists an x that satisfies the predicate, it is guaranteed that there exists at least one x value that satisfies the predicate for all y values. Hence, if ∀y ∃x P(x, y) is true, then ∃x ∀y P(x, y) is also true.
The statements ∃x ∀y P(x, y) and ∀y ∃x P(x, y) are equivalent. Swapping the order of the quantifiers does not change the overall meaning of the statement.
Both statements assert the existence of an x value that satisfies a predicate for all possible y values, albeit with a different syntactic structure.
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