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 $X=Y$?
Union and Intersection Proof
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elementary-set-theory
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0Give yourself two sets and see if this is true in general – 2011-10-19
1 Answers
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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 $X = Y$.