Question

Determine Poset properties of the given relation.

Short answer

\(R\) is not a poset, as it does not satisfy any of the properties.

Step-by-step solution

Given data

\(R\) is the relation, \(a\) related to \(b\) if \(a\) is not equal to \(b\).

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 not reflexive as a cannot be unequal to \(a\).

\(R\) is not antisymmetric. \(R\) is not transitive, for distinct \(a\) and \(b\), \(a\) is unequal to \(b\), \(b\) is unequal to \(a\), but \(a\) is not unequal to \(a\). \((R,\quad \ne )\) is not a poset, as it does not satisfy any or the properties.