Are the members of (x,y) if and only if (y = x + 3) & (y = x – 3) an empty set?

Hey Bob4040 and welcome to the forums.

Your definition is a little vague. You mention integers but then you say x and y are real numbers which is a bit of a contradiction.

Are you trying to ask whether x and y are real numbers if the relation holds?

 

Ohh I see what you mean now.

Well the easiest way to check is to set them equal to one another and see if you get a solution or a contradiction.

y = x + 3, y = x - 3 implies x + 3 = x - 3 which implies 3 = -3 which is a contradiction, so you have proven that no element exists satisfying the relationship so you have the empty set.

 

I know this is not what you meant/intended , Bob4040, but, strictly speaking your

identity would hold if x were in the set called Z_3, where 3=-3 :

http://www.wolframalpha.com/input/?i=Z+mod+3

Then,for all x, x+3=x-3 .