Question

Can a relation on a set be neither reflexive nor irreflexive?

Short answer

For example, \(A = \{ 1,2,3,4\} \) has set as \(\{ (1,1),(1,2),(3,4),(4,4)\} \) which is neither reflexive nor irreflexive.

Step-by-step solution

Given data

Consider a relation in a set.

Concept used of relation

A relation\({\rm{R}}\)on a set\({\rm{A}}\)is called reflexive if\((a,a) \in R\)for every element\(a \in A\).

A relation\({\rm{R}}\)on a set\({\rm{A}}\)is called symmetric if\((b,a) \in R\)whenever\((a,b) \in R\), for all\(a,b \in A\)

A relation\({\rm{R}}\)on a set\({\rm{A}}\)such that for all\(a,b \in A\), if\((a,b) \in R\)and\((b,a) \in R\)then\(a = b\)is called anti symmetric.

A relation\({\rm{R}}\)on a set\({\rm{A}}\)is called transitive if whenever\((a,b) \in R\)and\((b,c) \in R\)then\((a,c) \in R\)for all\(a,b,c \in A\)

Solve for relation

Let \(R\) be a binary relation on a set \(A\).

\({\rm{R}}\)is neither reflexive nor irreflexive if there is \(x \in A\) such that \((x,x) \in R\) and there is \(y \in A\) such that \((y,y) \notin R\)

For example, \(A = \{ 1,2,3,4\} \) has set as \(\{ (1,1),(1,2),(3,4),(4,4)\} \) which is neither reflexive nor irreflexive.