Answer to Question #267845 in Discrete Mathematics for Moon

Question

Show that A ⊕ B = (A ∪ B) - (A ∩ B).

Answer 1

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

 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