# Definition:Strictly Precede

## Definition

### Definition 1

Let $\left({S, \prec}\right)$ be a strictly ordered set.

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

Then $a$ strictly precedes $b$.

### Definition 2

Let $\left({S, \preceq}\right)$ 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 $\left({S, <}\right)$ 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.