Equivalence of Definitions of Symmetric Difference/(1) iff (2)

From ProofWiki
Jump to navigation Jump to search

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

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


Proof

\(\ds S \symdif T\) \(=\) \(\ds \paren {S \setminus T} \cup \paren {T \setminus S}\) 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

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Equivalence_of_Definitions_of_Symmetric_Difference/(1)_iff_(2)&oldid=521724"