Show that A ⊕ B = (A ∪ B) - (A ∩ B).

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$A \oplus B=\{x \mid x \in A \oplus B\}$

By the definition of symmetric difference $A \oplus B$ , $x$ then has to be an element of A or an element of B, but not an element of both.

$=\{x \mid(x \in A \vee x \in B) \wedge \neg(x \in A \wedge x \in B)\}$

By the definition of the union:

$=\{x \mid(x \in A \cup B) \wedge \neg(x \in A \wedge x \in B)\}$

By the definition of the intersection:

$=\{x \mid(x \in A \cup B) \wedge \neg(x \in A \cap B)\}$

By the definition of the difference:

$\begin{gathered} =\{x \mid x \in(A \cup B)-(A \cap B)\} \\ =(A \cup B)-(A \cap B) \end{gathered}$

Hence Proved