# Dual of Preordered Set is Preordered Set

Jump to navigation
Jump to search

## Theorem

Let $P = \struct {S, \preceq}$ be a preordered set.

Then dual of $P$, $P^{-1} = \struct {S, \succeq}$ is also a preordered set.

This needs considerable tedious hard slog to complete it.In particular: Dual Ordered Set $\ne$ Dual Preordered SetTo discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Proof

By Inverse of Reflexive Relation is Reflexive:

- $\succeq$ is reflexive.

By Inverse of Transitive Relation is Transitive:

- $\succeq$ is transitive.

Hence $\succeq$ is a preordering.

$\blacksquare$

## Sources

- Mizar article YELLOW_7:4

- Mizar article YELLOW_7:6