# Definition:Precede

(Redirected from Definition:Predecessor)

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$ such that $a \preceq b$.

Then $a$ precedes $b$.

## Also known as

The statement $b$ precedes $a$ can be expressed as $b$ is a predecessor of $a$.

If it is important to make the distinction between a predecessor and a strict predecessor, the term weak predecessor can be used for predecessor.

When the underlying set $S$ of the ordered set $\left({S, \leqslant}\right)$ is one of the sets of numbers $\N$, $\Z$, $\Q$, $\R$ or a subset, the term is less than or equal to is usually used instead of precedes.

## Also defined as

Some sources use the term predecessor to mean immediate predecessor.