# Order Relations

Learn about order relations, the usual order, and the dual order.

## We'll cover the following

## Order relation

A relation **order relation** if

For any

$a\in A$ , there is$aRa$ . This means that$R$ is reflexive.For any

$a,b \in A$ , if$aRb$ and$bRa$ , then$a=b$ . This means that$R$ is antisymmetric.For any

$a,b,c \in A$ , if$aRb$ and$bRc$ , then$aRc$ . This means that$R$ is transitive.

The relation **partial order,** and the set **ordered set** is the term used to describe a set

Note:We will interchangeably use both terms, i.e., order relation and partial order, to become comfortable with using both terms.

For a set *precedes* *succeeds*

### Examples

The subset relation is a partial order for any set of the sets. Let

As every set is a subset of itself, this means that

$\subseteq$ is a reflexive relation.For any sets

$A, B \in \cal C$ , if$A\subseteq B$ and$B\subseteq A$ , then$A=B$ . This means that$\subseteq$ is an antisymmetric relation.For any sets

$A, B, C \in \cal C$ , if$A\subseteq B$ and$B\subseteq C$ , then$A\subseteq C$ . This means that$\subseteq$ is a transitive relation.

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