# Definition:Upper Set

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## Definition

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

Let $U \subseteq S$.

### Definition 1

$U$ is an **upper set** in $S$ if and only if:

- $\forall u \in U: \forall s \in S: u \preceq s \implies s \in U$

### Definition 2

$U$ is an **upper set** in $S$ if and only if:

- $U^\succeq \subseteq U$

where $U^\succeq$ is the upper closure of $U$.

### Definition 3

$U$ is an **upper set** in $S$ if and only if:

- $U^\succeq = U$

where $U^\succeq$ is the upper closure of $U$.

## Also known as

An upper set is also known as an **upper-closed set** or **upward-closed set**. Some sources call it an **upset** or **up-set**.

## Also see

- Results about
**upper sets**can be found here.