# Equivalence of Definitions of Symmetric Difference/(3) 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 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.

$\blacksquare$