# Definition:Strictly Precede/Definition 2

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## Contents

## Definition

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**.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets