Inverse of Ordering is Ordering
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Theorem
If $\preceq$ is an ordering on $S$, then so is its inverse $\preceq^{-1}$.
The inverse of an ordering is usually denoted by reversing its symbol, thus $\preceq^{-1}$ is written $\succeq$.
Proof
By Inverse Relation Properties, if a relation is reflexive, transitive and/or antisymmetric, then so is its inverse.
The result follows.
$\blacksquare$
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$: Theorem $14.2$
