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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'symmetric difference'
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a suggestion by matte
Correction id: 4797
Filed on: 2004-09-15 03:15:37
Status: Accepted on 2004-09-18 12:06:15
Type: Addendum
Correction text:
Hi
Below are some comments on this entry that you could consider:
1) Create a list of the properties. Maybe something like:
\subsubsection*{Properties}
Suppose $A,B$ and $C$ are sets.
\begin{enumerate}
\item
\begin{eqnarray*}
A \Delta A &=& (A-B) \cup (B-A) \\
&=& (A \cup B) - (A \cap B), \\
A \Delta A&=&\emptyset, \\
A\Delta\emptyset&=&A, \\
A\cap(B\Delta C)&=&(A\cap B)\Delta(A\cap C), \\
(A\Delta B)\Delta C &=& A\Delta (B \Delta C).
\end{eqnarray*}
The last property shows that the mapping $(A,B)\mapsto A\Delta B$ is
associative.
\item
In general, an element will be in the symmetric difference of
several sets if and only if it is in an odd number of the sets.
\item
If $f\colon X\to Y$ and $A,B\subseteq Y$, then
f^{-1}(A \bigtriangleup B) = f^{-1}(A) \bigtriangleup f^{-1}(B)
where $f^{-1}$ is the inverse image.
\end{enumerate}
(The last property is not in the current entry.)
2) Move the (longer) proofs into separate entries(?)
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No comment from correction handler mathcam.
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