Solution to Show that A ⊕ B = (A ∪ B) - (A ∩ B). - Sikademy
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Archangel Macsika

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

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