Question

Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.

Short answer

\(\{ ({\rm{a}},{\rm{b}})\mid {\rm{a}}\) and \({\rm{b}}\) speak a common language \(\} \) is not an equivalence relation .

Step-by-step solution

Given data

Given data is \(\{ (a,b)\mid a\) and b speak a common language \(\} \).

Concept used of equivalence relation

An equivalence relation is a binary relation that is reflexive, symmetric and transitive.

Solve for relation

In order to determine if a relation is an equivalent relation, we need to see if said relation is a reflexive, symmetric and transitive.

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

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

A relation on a set \({\rm{A}}\) is transitive if \((a,b) \in R\) and \((b,c) \in R\) implies \((a,c) \in R\).

It is not transitive since if \(x\) and \(y\) speak a common language, say Spanish; and \(y\) and \(z\) speak another common language, say English; it doesn't mean that \(x\) and \(z\) speak a common language .