# Definition:Precede

(Redirected from Definition:Predecessor)

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## 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.

## Also see

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order