Question

To find the smallest relation of the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) which is reflexive, symmetric and transitive.

Short answer

The smallest relation of the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) which is reflexive, symmetric and transitive.

\(\{ (1,1),(2,2),(3,3),(4,4),(1,2)(1,4),(2,4)(2,1),(4,2),(4,1)\} \).

Step-by-step solution

Given data

The relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \).

Concept used of types 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\({\rm{A}}\)is called transitive if whenever\((a,b) \in R\)and\((b,c) \in R\)then\((a,c) \in R\)for all\(a,b,c \in A\).

Find the smallest relation

Consider the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) defined on set \(A = \{ 1,2,3,4\} \).

The relation

\(\{ (1,1),(2,2),(3,3),(4,4),(1,2)(1,4),(2,4)(2,1),(4,2),(4,1)\} \)

Is the smallest relation which is Reflexive.

\(\{ (1,1),(2,2),(3,3),(4,4)\} \in \{ 1,2,3,4\} \)

Symmetric,

\((1,2)(2,1) \in R\)

\((4,2),(2,4) \in R\)

and transitive relation,

\((4,1)\} (1,4) \in R\)

\((1,1),(2,2) \in R{\rm{ }}(1,2) \in R\),

\((1,2)(1,4) \in R{\rm{ }}(2,4) \in R\)

Therefore,

The smallest relation of the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) which is reflexive, symmetric and transitive is \(\{ (1,1),(2,2),(3,3),(4,4),(1,2)(1,4),(2,4)(2,1),(4,2),(4,1)\} \).