Question

Define three equivalence relations on the set of classes offered at your school. Determine the equivalence classes for each of these equivalence relations.

Short answer

The three equivalence relations are,

\({(a)_{{R_1}}} = \{ b \in A\mid b\)is a Math class \(\} \)

\({(a)_{{R_2}}} = \{ b \in A\mid b\)has the same number of students as \(a\} {\rm{ }}\)

\({(a)_{{R_3}}} = \{ b \in A\mid b\)has the same teacher as \(a\} {\rm{ }}\).

Step-by-step solution

Given data

Given data is equivalence relation.

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 \((a,c) \in R\).

 Step 4: Define three equivalence relations

\(A = \)Set of classes offered at your school.

We define to three equivalence relations.

For example:

\(\begin{array}{l}{R_1} = \{ (a,b)\mid a{\rm{ and }}b{\rm{ are both Math classes }}\} \\{R_2} = \{ (a,b)\mid a{\rm{ and }}b{\rm{ have the same number of students }}\} \\{R_3} = \{ (a,b)\mid a{\rm{ and }}b{\rm{ have same teacher }}\} \end{array}\)

Note: If the statements states that a andb 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 \(a\).

\(\begin{array}{l}{(a)_{{R_1}}} = \{ b \in A\mid b{\rm{ is a Math class }}\} \\{(a)_{{R_2}}} = \{ b \in A\mid b{\rm{ has the same number of students as }}a\} \\{(a)_{{R_3}}} = \{ b \in A\mid b{\rm{ has the same teacher as }}a\} \end{array}\)