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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 $X=Y$?

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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
up vote 4 down vote accepted

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 $X = Y$.

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