Union and Intersection Proof

Question

Say we have two sets of real numbers, X and Y. Say that $X\cup Y=X\cap Y$. Is it true to say that $X=Y$?

Answer 1

It is true for any set.

First, $$X \cup Y = X \cap Y \subseteq X,$$ so $Y \subseteq X$. (Basically, the line above says that taking the union with $Y$ adds nothing that $X$ did not already possess, so $Y$ must be a subset of $X$.)

Similarly, $$X \cup Y = X \cap Y \subseteq Y,$$ so $X \subseteq Y$.

Together, these observations mean that $X = Y$.