Equivalence of Definitions of Symmetric Difference

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 2

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

Definition 3

$S * T = \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)$

Definition 4

$S * T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$

Definition 5

$S * T := \set {x: x \in S \oplus x \in T}$

Proof

$(1)$ iff $(2)$

 $\displaystyle S * T$ $=$ $\displaystyle \paren {S \setminus T} \cup \paren {T \setminus S}$ Definition 1 of Symmetric Difference $\displaystyle$ $=$ $\displaystyle \paren {\paren {S \cup T} \setminus T} \cup \paren {\paren {S \cup T} \setminus S}$ Set Difference with Union is Set Difference $\displaystyle$ $=$ $\displaystyle \paren {S \cup T} \setminus \paren {T \cap S}$ De Morgan's Laws: Difference with Intersection $\displaystyle$ $=$ $\displaystyle \paren {S \cup T} \setminus \paren {S \cap T}$ Intersection is Commutative

$\Box$

$(1)$ iff $(3)$

 $\displaystyle S * T$ $=$ $\displaystyle \paren {S \setminus T} \cup \paren {T \setminus S}$ Definition 1 of Symmetric Difference $\displaystyle$ $=$ $\displaystyle \paren {S \cap \overline T} \cup \paren {\overline S \cap T}$ Set Difference as Intersection with Complement

$\Box$

$(2)$ iff $(4)$

 $\displaystyle S * T$ $=$ $\displaystyle \left({S \cup T}\right) \setminus \left({S \cap T}\right)$ Symmetric Difference: Definition 2 $\displaystyle$ $=$ $\displaystyle \left({S \cup T}\right) \cap \left({\overline {S \cap T} }\right)$ Set Difference as Intersection with Complement $\displaystyle$ $=$ $\displaystyle \left({S \cup T}\right) \cap \left({\overline S \cup \overline T}\right)$ De Morgan's Laws: Complement of Intersection

$\Box$

$(2)$ iff $(5)$

 $\displaystyle x \in S * T$ $\iff$ $\displaystyle x \in S \oplus x \in T$ Symmetric Difference: Definition 5 $\displaystyle$ $\iff$ $\displaystyle \left({x \in S \lor x \in T} \right) \land \neg \left({x \in S \land x \in T}\right)$ Definition of Exclusive Or $\displaystyle$ $\iff$ $\displaystyle \left({x \in S \cup T}\right) \land \left({x \notin S \cap T}\right)$ Definition of Set Intersection and Set Union $\displaystyle$ $\iff$ $\displaystyle x \in \left({S \cup T}\right) \setminus \left({S \cap T}\right)$ Definition of Set Difference

The result follows by definition of set equality.

$\Box$

$(3)$ iff $(5)$

 $\displaystyle x \in S * T$ $\iff$ $\displaystyle x \in S \oplus x \in T$ Symmetric Difference: Definition 5 $\displaystyle$ $\iff$ $\displaystyle \left({\neg \left({x \in S}\right) \land \left({x \in T}\right)}\right) \lor \left({\left({x \in S}\right) \land \neg \left({x \in T}\right)}\right)$ Non-Equivalence as Disjunction of Conjunctions $\displaystyle$ $\iff$ $\displaystyle \left({x \in \overline S \land x \in T}\right) \lor \left({x \in S \land x \in \overline T}\right)$ Definition of Set Complement $\displaystyle$ $\iff$ $\displaystyle \left({x \in \overline S \cup T}\right) \lor \left({x \in S \cup \overline T}\right)$ Definition of Set Intersection $\displaystyle$ $\iff$ $\displaystyle x \in \left({\overline S \cup T}\right) \cup \left({S \cup \overline T}\right)$ Definition of Set Union $\displaystyle$ $\iff$ $\displaystyle x \in \left({S \cup \overline T}\right) \cup \left({\overline S \cup T}\right)$ Union is Commutative

The result follows by definition of set equality.

$\blacksquare$