Prove that X = Y where X is a solution to x^2 - 1 = 0 and y = 1,-1

Prove that X = Y where X is a solution to x^2 - 1 = 0 and y = 1,-1

This is a set theory proof. The idea is that if X is a subset of Y, and Y is a subset of X, then X = Y.

So, first we show that Y is a subset of X. And we can do this by simply substituting the values for y in x^2 - 1 = 0. We see that both 1 and -1 work when substituted for X, and so Y is a subset of X.

Now we have to prove that X is a subset of Y. If we factor x^2 - 1 we get (x-1)(x+1) = 0. It is evident that this product is 0 exactly when x=1 or x=-1; so X is a subset of Y.

Now since Y is a subset of X, and X is a subset of Y, then X=Y