= Solution
Use <De Morgan's laws> to remove the <complement of a set>, then distribute the two <unions>:
$$
\begin{aligned}
(A\cup B)\setminus(A\cap B)
&=(A\cup B)\cap(\overline A\cup\overline B)\\
&=(A\cap\overline A)\cup(A\cap\overline B)\cup(B\cap\overline A)\cup(B\cap\overline B)\\
&=(A\cap\overline B)\cup(B\cap\overline A).
\end{aligned}
$$
The first and last <intersections> in the middle line are the <empty set>. The remaining <union> is precisely the collection of elements belonging to exactly one of the original <sets>. Therefore both expressions define the same <symmetric difference>:
$$
\boxed{(A\cup B)\setminus(A\cap B)=(A\setminus B)\cup(B\setminus A)=A\oplus B.}
$$
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