# Characterization of Minimal Element

Jump to navigation
Jump to search

This article needs proofreading.Please check it for mathematical errors.If you believe there are none, please remove `{{Proofread}}` from the code.To 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 `{{Proofread}}` from the code. |

This article needs to be linked to other articles.including categoryYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To 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 `{{MissingLinks}}` from the code. |

## Theorem

Let $C$ be a class.

Let $\prec$ be a relation on $C$.

Let $B$ be a subclass of $C$.

Let $x \in B$.

Let $S_x = \set {y \in C: y \prec x \text{ and } y \ne x}$ be the initial segment of $x$ in $C$.

Then $x$ is a minimal element of $B$ if and only if $B \cap S_x = \O$.

## Proof

### Necessary Condition

Suppose $x$ is a minimal element of $B$.

Then for each $z \in B$ such that $z \ne x$, $z \nprec x$.

Thus $S_x \cap B = \O$.

$\Box$

### Sufficient Condition

Suppose that $x$ is not a minimal element of $B$.

Then for some $z \in B$, $z \prec x$ and $z \ne x$.

Thus $z \in S_x$.

Since $z \in B$, $B \cap S_x \ne \O$.

$\blacksquare$