Equivalence of Definitions of Symmetric Difference/(2) iff (5)

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Theorem

Let $S$ and $T$ be sets.

The following definitions of the concept of symmetric difference $S * T$ between $S$ and $T$ are equivalent:

Definition 2

$S * T = \paren {S \cup T} \setminus \paren {S \cap T}$

Definition 5

$S * T := \set {x: x \in S \oplus x \in T}$


Proof

\(\displaystyle x \in S * T\) \(\iff\) \(\displaystyle x \in S \oplus x \in T\) Symmetric Difference: Definition 5
\(\displaystyle \) \(\iff\) \(\displaystyle \left({x \in S \lor x \in T} \right) \land \neg \left({x \in S \land x \in T}\right)\) Definition of Exclusive Or
\(\displaystyle \) \(\iff\) \(\displaystyle \left({x \in S \cup T}\right) \land \left({x \notin S \cap T}\right)\) Definition of Set Intersection and Set Union
\(\displaystyle \) \(\iff\) \(\displaystyle x \in \left({S \cup T}\right) \setminus \left({S \cap T}\right)\) Definition of Set Difference

The result follows by definition of set equality.

$\blacksquare$