Question

Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. Determine the equivalence classes for each of these equivalence relations.

Short answer

The three equivalence relations are,

\({(a)_{{R_1}}} = \{ b\mid b\) has the same gender as \(a\} {\rm{ }}\)

\({(a)_{{R_2}}} = \{ b\mid b\) speaks German\(\} {\rm{ }}\)

\({(a)_{{R_3}}} = \{ b\mid b\) lives in same town as \(a\} {\rm{ }}\).

Step-by-step solution

Given data

Given data is equivalence relation.

 Step 2: Concept used of equivalence relation

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

Show the equivalence 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 \(A\) 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 \((a,c) \in R\).

\(A = \) Set of students in your discrete mathematics class.

Define three equivalence relations

We define to three equivalence relations.

For example:

\({R_1} = \{ (a,b)\mid a\) and \(b\) have same gender \(\} \)

\({R_2} = \{ (a,b)\mid a\) and \(b\) both speak German \(\} \)

\({R_3} = \{ (a,b)\mid a\) and \(b\) live in same town \(\} \)

Note: If the statements states that a and b have the same property, then the relation tends to be an equivalence relation.

The equivalence class of \(a\) is the set of all elements that are in relation to \(a\).

\({(a)_{{R_1}}} = \{ b\mid b\) has the same gender as \(a\} \)

\({(a)_{{R_2}}} = \{ b\mid b\) speaks German\(\} \)

\({(a)_{{R_3}}} = \{ b\mid b\) lives in same town as \(a\} \)