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

Bob4040
Is the set of all integers where <x,y> $\in$ R 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?

Bob4040
I do not mean all real numbers. R is the set. x and y are members of the set on the condition that (y = x + 3) & (y = x – 3). These two lines never intersect, so I think that means it is an empty set?

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.

Bob4040
Thank you for the help!