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

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.

$\blacksquare$