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

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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 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

$\blacksquare$


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