Question

Determine Poset properties of the given relation.

Short answer

\(R\) is not a poset, as it is not antisymmetric.

Step-by-step solution

Given data

\(R\) consists of the ordered pairs \(\{ (0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(2,0),(2,2),(3,3)\} \).

Concept used of partially ordered set

A relation\(R\)is a poset if and only if,\((x,x)\)is in\({\rm{R}}\)for all x (reflexivity)

\((x,y)\)and\((y,x)\)in R implies\(x = y\)(anti-symmetry),\((x,y)\)and\((y,z)\)in R implies\((x,z)\)is in\({\rm{R}}\)(transitivity).

Find  if the \(R\) is poset

\(R\)is reflexive as every element is related to itself.

\(R\)is not antisymmetric as \((0,1)\) and \((1,0)\) both belong to \(R\).

\(R\)is transitive.

So, \(R\) is not a poset.