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

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Theorem

Let $S$ and $T$ be sets.

The following definitions of the concept of symmetric difference $S \symdif T$ between $S$ and $T$ are equivalent:

Definition 2

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

Definition 4

$S \symdif T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$


Proof

\(\ds S \symdif T\) \(=\) \(\ds \paren {S \cup T} \setminus \paren {S \cap T}\) Definition 2 of Symmetric Difference
\(\ds \) \(=\) \(\ds \paren {S \cup T} \cap \paren {\overline {S \cap T} }\) Set Difference as Intersection with Complement
\(\ds \) \(=\) \(\ds \paren {S \cup T} \cap \paren {\overline S \cup \overline T}\) De Morgan's Laws: Complement of Intersection

$\blacksquare$