# Definition:Strictly Precede

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## Definition

### Definition 1

Let $\struct {S, \prec}$ be a strictly ordered set.

Let $a, b \in S$ and $a \prec b$.

Then **$a$ strictly precedes $b$**.

### Definition 2

Let $\struct {S, \preceq}$ be an ordered set.

Let $a \preceq b$ such that $a \ne b$.

Then **$a$ strictly precedes $b$**.

## Notation

When $a \preceq b$ and $a \ne b$, it is usual to denote this with the symbol:

- $a \prec b$

and similar derived notation for other ordering symbols.

## Also known as

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

Some sources refer to a **strict predecessor** simply as a **predecessor**.

When the underlying set $S$ of the ordered set $\struct {S, <}$ is one of the sets of numbers $\N$, $\Z$, $\Q$, $\R$ or a subset, the term **is less than** is usually used instead of **(strictly) precedes**.

## Also see

- Reflexive Reduction of Ordering is Strict Ordering where it is demonstrated that the $\prec$ relation can be defined as the reflexive reduction of the $\preceq$ relation.