Question

Describe Partially Ordered Set.

Answer 1

To know what a partially ordered set is we first have to know what a set is. A set is a well defined collection of objects or symbols which are known as the members or the elements of the set. The elements of a set have the same properties.

Complete step by step solution:
We have to describe what a Partially Ordered Set is.
A partially ordered set is that set which is taken together with a partial order. It is defined as an ordered pair $P=\left( X,\le \right)$ where we take $X$ as the ground set of $P$ and $\le$ is known as the partial order of $P$.
We can also define the partial order set as below:
Let us consider any relation $R$ on a set $S$ which satisfy the following properties:
1. $R$ is reflexive
If $xRx$ for every $x\in S$
2. $R$ is antisymmetric
If $xRy$ and $yRx$ then $x=y$
3. $R$ is transitive
If $xRy$ and $yRz$ then $xRz$
In this case $R$ is known as partial order relation and the set $S$ with the partial order is known as the Partial Order Set or we can say a POSET which we can denote by $\left( S,\le \right)$

Note: Some extra information about the partial order set is that an element $v$ in partial ordered set is said to be its upper bound for any subset $A$ of $X$ if, for every $a\in A$ we have $a\le v$. We can represent a POSET in a form of simple diagram which is known as Hasse diagram. Ordered pair is a pair of numbers such as $\left( x,y \right)$ which are written in a particular order we can say that ordered pair $\left( x,y \right)$ is not same as ordered pair $\left( y,x \right)$.