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.

15. b): There exists an x such that, for all y, x + y is even. This is the correct answer for this on Chegg and in the back of the book but when you look at the solution under 19. on Chegg they changed the wording to "for an x there exist all y such that x + y is even".

Why is the wording changed? Which is correct?

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.

How is this False?

Thanks

Explanation / Answer

This is false

Because an even +even =even

And odd+odd=even

So If x is chosen to be odd then y has to be odd to make x+y an even number.this means x+y is even for all y will not hold true because it will be true only when y is also odd.

Similarly Even +Even =Even ,So if x is already a given even number in that case x+y can be even only for an even y.It cant be even for all values of y.

Chegg shouldnt have changed the wording to "for an x there exist all y such that x+y is even". Because this statement is false.

Now come to original statement "there exist an x such that,for all y,x+y is even"

Thos statement is true because given any y it may either be even or odd.

If y is even you can find out any even x to make x+y even

And if y is odd you can find any odd x to make it even.

So original statement is true

But chegg changed version whose meaning is altogether different is false.

I hope that settles the issue.

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