# Equivalence of Definitions of Symmetric Difference/(2) iff (4)

## 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 4

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

## Proof

 $\displaystyle S * T$ $=$ $\displaystyle \left({S \cup T}\right) \setminus \left({S \cap T}\right)$ Symmetric Difference: Definition 2 $\displaystyle$ $=$ $\displaystyle \left({S \cup T}\right) \cap \left({\overline {S \cap T} }\right)$ Set Difference as Intersection with Complement $\displaystyle$ $=$ $\displaystyle \left({S \cup T}\right) \cap \left({\overline S \cup \overline T}\right)$ De Morgan's Laws: Complement of Intersection

$\blacksquare$