Equivalence of Definitions of Symmetric Difference/(2) 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 2

$S \symdif T = \paren {S \cup T} \setminus \paren {S \cap T}$

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

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {x \in S \lor x \in T} \land \neg \paren {x \in S \land x \in T}\)

Definition of Exclusive Or

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {x \in S \cup T} \land \paren {x \notin S \cap T}\)

Definition of Set Intersection and Definition of Set Union

\(\ds \)

\(\leadstoandfrom\)

\(\ds x \in \paren {S \cup T} \setminus \paren {S \cap T}\)

Definition of Set Difference

The result follows by definition of set equality.

$\blacksquare$