Symmetric Closure of Inverse Relation
Jump to navigation
Jump to search
Definition
Let $\RR$ be a relation on a set $S$.
Let $\RR^{-1}$ be the inverse of $\RR$.
Let $\RR^\leftrightarrow$ be the symmetric closure of $\RR$.
Then:
- $\RR^\leftrightarrow = \paren {\RR^{-1} }^\leftrightarrow$
Proof
\(\ds \RR^\leftrightarrow\) | \(=\) | \(\ds \RR \cup \RR^{-1}\) | Definition of Symmetric Closure | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\RR^{-1} }^{-1} \cup \RR^{-1}\) | Inverse of Inverse Relation | |||||||||||
\(\ds \) | \(=\) | \(\ds \RR^{-1} \cup \paren {\RR^{-1} }^{-1}\) | Union is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\RR^{-1} }^\leftrightarrow\) | Definition of Symmetric Closure |
$\blacksquare$