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 $S \symdif T$ between $S$ and $T$ are equivalent:

Definition 1

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

Definition 3

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

Proof

\(\ds S \symdif T\)

\(=\)

\(\ds \paren {S \setminus T} \cup \paren {T \setminus S}\)

Definition 1 of Symmetric Difference

\(\ds \)

\(=\)

\(\ds \paren {S \cap \overline T} \cup \paren {\overline S \cap T}\)

Set Difference as Intersection with Complement

$\blacksquare$