Definition:Precede

Definition

Let $\preceq$ be an ordering.

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

Then $a$ precedes $b$.

$a$ is then described as being a predecessor of $b$.

Also known as

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

If it is important to make the distinction between a predecessor and a strict predecessor, the term weak predecessor can be used for predecessor.

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

Also defined as

Some sources use the term predecessor to mean immediate predecessor.

Also see

• Definition:Strictly Precede
• Definition:Immediate Predecessor Element

• Definition:Succeed

• Results about predecessor elements can be found here.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order