Symmetric Closure/Examples/Arbitrary Example 1

Examples of Symmetric Closures
Let $S = \set {1, 2, 3, 4, 5}$ be a set.

Let $\RR$ be the relation on $S$ defined as:
 * $\RR = \set {\tuple {1, 2}, \tuple {2, 3}, \tuple {3, 4}, \tuple {5, 4} }$

The symmetric closure $\RR^\leftrightarrow$ of $\RR$ is given by:
 * $\RR^* = \set {\tuple {1, 2}, \tuple {2, 3}, \tuple {3, 4}, \tuple {5, 4}, \tuple {2, 1}, \tuple {3, 2}, \tuple {4, 3}, \tuple {4, 5} }$

Proof
By definition of symmetric closure:
 * $\RR^\leftrightarrow = \RR \cup \RR^{-1}$

where $\RR^{-1}$ denotes the inverse of $\RR$.

By definition of inverse relation:
 * $\RR^{-1} = \set {\tuple {2, 1}, \tuple {3, 2}, \tuple {4, 3}, \tuple {4, 5} }$

The result then follows by definition of set union.