In the book Discrete Mathematical Structures 6th edition, section 2.1... Questio

Question

In the book Discrete Mathematical Structures 6th edition, section 2.1...

Question 19. asks to determine the truth value of questions 14 - 18. I don't understand how 15. b) is False. The book, Chegg, and my professor all state it is False.

If it states in the exercise, P (x): x is even; Q (x): x is a prime number; R (x, y): x + y is even. The variables x and y represent integers.

Then, do I need to apply the P (x) and Q (x) statements when working with R (x, y) if the question only references R (x, y)?

15. b) there Exist x for All y R (x, y)

15. b) in plain English: There exists an x such that, for all y, x + y is even. —This is what the back of the book has for 15. b)—

19. for 15. b): False —This is what the back of the book, Chegg, and my professor state for 19. in reference to 15. b)—

Maybe this is what I am doing wrong. I am NOT considering P (x) or Q (x) for 15. b) since it doesn’t reference those other two statements. I am only looking at R (x, y): x + y is even. The variables x and y represent integers.

Couldn't I always find some x (there exists an x) for each and any y (for all y) that would add to an even number for x + y?

i.e. if y = 3, I could find x = 1 and have an even number, 1 + 3 = 4 and so on.

The book, Chegg, and my professor all state it is False but I still don't understand why. Is it because I do still HAVE to apply the P (x) and Q (x) statements even when the question is in reference to R (x, y)?

I asked this question previously on Chegg and the answer was given to me as True which is what I think it should be. But I am being told that is incorrect and it should be False.

How is this False?

Also, if 15. b) is False then how is 15. a) True. Wouldn’t the same rule apply? --This True answer for 15. a) is verified as well.--

15. a) for All x there Exists y R(x, y)

Thanks

Explanation / Answer

P (x): x is even; Q (x): x is a prime number; R (x, y): x + y is even

15. b) x y R (x, y)

This statement is FALSE.

Correct way of reading: There exists an x such that for all y, x + y is even. (As given in the book)

Wrong way of reading: For all y, there exists an integer x such that x + y is even.

That statement would be y x R (x, y)

Proof why the statement is false: The x should be such that x + y is even for ALL y.

Please note, the x must be the SAME for ALL y.

This means it x + y is even if y is odd => x i odd as the sum of two odds is even.

The x should be such that x + y is even for ALL x.

This means it x + y is even if y is even => x i even as the sum of two evens is even.

The two statements contradict each other, x can't be both even and odd and the same time.

Thus the statement is false.

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