Equivalence of Definitions of Symmetric Difference/(3) iff (5)
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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 3
- $S \symdif T = \paren {S \cap \overline T} \cup \paren {\overline S \cap T}$
Definition 5
- $S \symdif T := \set {x: x \in S \oplus x \in T}$
Proof
\(\ds \) | \(\) | \(\ds x \in S \symdif T\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in S \oplus x \in T\) | Definition 5 of Symmetric Difference | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {\neg \paren {x \in S} \land \paren {x \in T} } \lor \paren {\paren {x \in S} \land \neg \paren {x \in T} }\) | Non-Equivalence as Disjunction of Conjunctions | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in \overline S \land x \in T} \lor \paren {x \in S \land x \in \overline T}\) | Definition of Set Complement | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in \overline S \cup T} \lor \paren {x \in S \cup \overline T}\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {\overline S \cup T} \cup \paren {S \cup \overline T}\) | Definition of Set Union | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {S \cup \overline T} \cup \paren {\overline S \cup T}\) | Union is Commutative |
The result follows by definition of set equality.
$\blacksquare$