Step 1

The relation is reflexive, since \(\displaystyle{\left({a},{a}\right)}\in{R}\), where \(\displaystyle{a}\in{A}\)

Also, since \(\displaystyle{a}={b}\) where \(\displaystyle{\left({a},{b}\right)}\in{R}\) therefore, the relation is symmetric.

Step 2

The relation is trivial transitive, as

\(\displaystyle{\left({a},{b}\right)}\in{R},{\left({b},{c}\right)}\in{R},\rightarrow{\left({a},{c}\right)}\in{R}\)

\(\displaystyle{a}={b}={c}\)

The relation is reflexive, symmetric and transitive.

therefore, r is an equivalence relation.

The relation is reflexive, since \(\displaystyle{\left({a},{a}\right)}\in{R}\), where \(\displaystyle{a}\in{A}\)

Also, since \(\displaystyle{a}={b}\) where \(\displaystyle{\left({a},{b}\right)}\in{R}\) therefore, the relation is symmetric.

Step 2

The relation is trivial transitive, as

\(\displaystyle{\left({a},{b}\right)}\in{R},{\left({b},{c}\right)}\in{R},\rightarrow{\left({a},{c}\right)}\in{R}\)

\(\displaystyle{a}={b}={c}\)

The relation is reflexive, symmetric and transitive.

therefore, r is an equivalence relation.