Equivalence of Definitions of Symmetric Difference/(3) iff (5)

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.

$\blacksquare$