Let P be a predicate. Determine whether or not each of the following implications is true and give a brief English explanation for your answer. 1) Vx=yP (x, y) + 3yVxP (x, y) 2) ByVxP (x, y) + VxJyP (x, y)

Question

Mathematics

Isaac Stafford

23 August, 09:57

Let P be a predicate. Determine whether or not each of the following implications is true and give a brief English explanation for your answer. 1) Vx=yP (x, y) + 3yVxP (x, y) 2) ByVxP (x, y) + VxJyP (x, y)

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Answers (1)

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Wilson Villa · Answer 1

See answer below

Step-by-step explanation:

1), Probably, your implication is ∀x∃yP (x, y) ⇒∃x∀yP (x, y).

This implication is false. Consider the predicate P (x, y) : ="x
A short explanation would be, even if for all elements you can find one that makes a predicate true, you can not find one element that makes the predicate true for all elements.

2) Again, I assume that the predicate is ∃x∀yP (x, y) ⇒∀x∃yP (x, y)

This implication is false. Consider the predicate P (x, y) : ="x is integer and xy=0 " for real numbers x, y. Then ∃x∀yP (x, y) is true, we need to take x=0. However, ∀x∃yP (x, y) is false, if you take x=1/2, 1/2 will never be an integer, no matter the value of y.

A short explanation would be, even if you can find one element that makes a predicate true for all elements, you can not always take an arbitrary element and find some element that makes the predicate true.