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 |
$\blacksquare$