# Definition:Succeed

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## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

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

Then **$b$ succeeds $a$**.

## Also known as

The statement **$b$ succeeds $a$** can be expressed as **$b$ is a succcessor of $a$**.

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

When the underlying set $S$ of the ordered set $\left({S, \leqslant}\right)$ is one of the sets of numbers $\N$, $\Z$, $\Q$, $\R$ or a subset, the term **is greater than or equal to** is usually used instead of **succeeds**.

## Also defined as

Some sources use the term **successor** to mean immediate successor.