Definition:Strictly Precede/Definition 2

Definition

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.

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