Equivalence of Definitions of Symmetric Difference/(3) 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 3

$S * T = \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)$

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
\(\displaystyle \) \(\iff\) \(\displaystyle \left({\neg \left({x \in S}\right) \land \left({x \in T}\right)}\right) \lor \left({\left({x \in S}\right) \land \neg \left({x \in T}\right)}\right)\) Non-Equivalence as Disjunction of Conjunctions
\(\displaystyle \) \(\iff\) \(\displaystyle \left({x \in \overline S \land x \in T}\right) \lor \left({x \in S \land x \in \overline T}\right)\) Definition of Set Complement
\(\displaystyle \) \(\iff\) \(\displaystyle \left({x \in \overline S \cup T}\right) \lor \left({x \in S \cup \overline T}\right)\) Definition of Set Intersection
\(\displaystyle \) \(\iff\) \(\displaystyle x \in \left({\overline S \cup T}\right) \cup \left({S \cup \overline T}\right)\) Definition of Set Union
\(\displaystyle \) \(\iff\) \(\displaystyle x \in \left({S \cup \overline T}\right) \cup \left({\overline S \cup T}\right)\) Union is Commutative

The result follows by definition of set equality.

$\blacksquare$