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 |
$\blacksquare$