Question

For the following open sentence, the universe of discourse is N, the set of natural numbers. What does the statement about the open sentence mean? Also state whether the statement is true or false:

Open Sentence:                                        T(x, y) : xy is even.

Statement:                                                    ∃!x∀y T(x, y).

There exist an x such that for all y,  the product xy is even. (False)

For all  x and y, the product xy is even. (True)

For all  y , there exist a unique  x, such the product xy is even. (False)

There exist  x and y, such that the product xy is even. (True)

None of the above

Answer 1

What does the statement about the open sentence mean?Also state whether the statement is true or false:

Answer :- The correct answer is , None of the above

Explanation:   

The correct meaning of the statement about the open sentence is: For all  y , there exist a unique  x, such the product xy is even.

For an example , take x=2 , so when y= 1, xy = 2*1 =2

y=2, xy = 2*2=4

y=3, xy = 2*3=6 and so on.

So the statement that For all  y , there exist a unique  x, such the product xy is even is actually TRUE. But , it is gven as false. So we have choosen none of the above as a correct answer.