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

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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

$\blacksquare$