# Equivalence of Definitions of Symmetric Difference/(1) iff (2)

## 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 1

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

### Definition 2

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

## Proof

 $\displaystyle S * T$ $=$ $\displaystyle \paren {S \setminus T} \cup \paren {T \setminus S}$ Definition 1 of Symmetric Difference $\displaystyle$ $=$ $\displaystyle \paren {\paren {S \cup T} \setminus T} \cup \paren {\paren {S \cup T} \setminus S}$ Set Difference with Union is Set Difference $\displaystyle$ $=$ $\displaystyle \paren {S \cup T} \setminus \paren {T \cap S}$ De Morgan's Laws: Difference with Intersection $\displaystyle$ $=$ $\displaystyle \paren {S \cup T} \setminus \paren {S \cap T}$ Intersection is Commutative

$\blacksquare$