Question

Find the smallest equivalence relation on the set \(\{ a,b,c,d,e\} \) containing the relation \(\{ (a,b),(a,c),(d,e)\} \).

Short answer

The smallest equivalence relation is \(\left\{ {\begin{array}{*{20}{l}}{(a,a),(a,b),(b,b),(b,a),(a,c),(c,c),}\\{(c,a),(b,c),(c,b),(d,d),(d,e),(e,d),(e,e)}\end{array}} \right\}\).

Step-by-step solution

Given data

The set is \(\{ a,b,c,d,e\} \) and contain the relation \(\{ (a,b),(a,c),(d,e)\} \).

Concept used of equivalence relation

A relation defined on a set\(S\)is known as an equivalence relation if set is reflexive, symmetric and transitive.

Find the equivalence relation

First, make the relation symmetric.

\(\{ (a,b),(b,a),(a,c),(c,a),(d,e),(e,d)\} \)

Now, make the relation transitive.

\(\{ (a,b),(b,a),(a,c),(c,a),(b,c),(c,b),(d,e),(e,d)\} \)

These things are not transitive, because the relation is going \(a\) to \(b\) and \(b\) to \(a\) not yet \(a\) to \(a\).

Thus, the equivalence relation becomes:

\(\left\{ {\begin{array}{*{20}{l}}{(a,a),(a,b),(b,b),(b,a),(a,c),(c,c),}\\{(c,a),(b,c),(c,b),(d,d),(d,e),(e,d),(e,e)}\end{array}} \right\}\)

Hence, this is the smallest relation that contains the relation \(\{ (a,b),(a,c),(d,e)\} \).