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If $A \bigcup B = A \bigcup C$ then does $B = C$ ? I was thinking that it would be false because if $B$ is a subset of $A$, and $C$ is a subset of $A$, then $B \neq C$.

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    @Tim Cooper: You know one "why" intuitively. At the technical level, however, what you wrote is not quite right, after all we could have $B=C$. When something is false in general, one shows this by producing an *explicit* counterexample.2011-10-10

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You're right.

An easy counterexample: $A=\{0,1\}$, $B=\{0\}$ and $C=\{1\}$.

You have $A\cup B=A\cup C=A$, but $B\ne C$.