Equivalence of Definitions of Symmetric Difference/(1) iff (3)

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 3

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

Proof

\(\displaystyle S * T\)

\(=\)

\(\displaystyle \left({S \setminus T}\right) \cup \left({T \setminus S}\right)\)

Symmetric Difference: Definition 1

\(\displaystyle \)

\(=\)

\(\displaystyle \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)\)

Set Difference as Intersection with Complement

$\blacksquare$