Definition:Strictly Precede

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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 $b$.


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