# Union and Intersection Proof

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 $X=Y$?

-
It is true for any set –  Yuan Oct 19 '11 at 3:15
Give yourself two sets and see if this is true in general –  smanoos Oct 19 '11 at 3:16

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 $X = Y$.