If we define symmetric difference as $A\triangle B=(A\setminus B)\cup(B\setminus A)$, can we also say that $(A\cup B)\triangle C=(A\triangle C)\cup(B\triangle C)$?
Is this symmetric difference statement valid?
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elementary-set-theory
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3If $x$ is in $B$ and $C$ but not in $A$, then it's not in the first set but it is in the second. – 2011-11-16
1 Answers
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Following Gerry's comment, here is an explicit example of how the equality can fail.
Let $A = \emptyset$ and $B = C = \{x\}$. We have $ \begin{align*} (A \cup B) \Delta C &= \{x\} \Delta \{x\}\\ &= \emptyset, \end{align*} $ while $ \begin{align*} (A \Delta C) \cup (B \Delta C) &= \{x\} \cup \emptyset\\ &= \{x\}. \end{align*} $