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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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    Give yourself two sets and see if this is true in general2011-10-19

1 Answers 1

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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 $X = Y$.