# Equivalence of Definitions of Symmetric Difference

## 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 1

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

### Definition 2

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

### Definition 3

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

### Definition 4

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

### Definition 5

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

## Proof

### $(1)$ iff $(2)$

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

$\Box$

### $(1)$ iff $(3)$

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

$\Box$

### $(2)$ iff $(4)$

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

$\Box$

### $(2)$ iff $(5)$

 $\ds$  $\ds x \in S \symdif T$ $\ds$ $\leadstoandfrom$ $\ds x \in S \oplus x \in T$ 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.

$\Box$

### $(3)$ iff $(5)$

 $\ds$  $\ds x \in S \symdif T$ $\ds$ $\leadstoandfrom$ $\ds x \in S \oplus x \in T$ 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.

$\blacksquare$