Equivalence of Definitions of Symmetric Difference/(2) iff (4)

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ and $T$ be sets.

The following definitions of the concept of symmetric difference $S * T$ between $S$ and $T$ are equivalent:

Definition 2

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

Definition 4

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


Proof

\(\displaystyle S * T\) \(=\) \(\displaystyle \left({S \cup T}\right) \setminus \left({S \cap T}\right)\) 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

$\blacksquare$