Question

Define three equivalence relations on the set of buildings on a college campus and 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 number of rooms as \(a\} {\rm{ }}\)

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

\({(a)_{{R_3}}} = \{ b\mid b\)has the same number of windows 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 \((a,c) \in R\).

Define three equivalence relations

\(A = \)Set of buildings on a college campus.

We define to three equivalence relations.

For example:

\({R_1} = \{ (a,b)\mid a\)and\(b\) have same number of rooms \(\} \)

\({R_2} = \{ (a,b)\mid a\)and\(b\) have same number of floors \(\} \)

\({R_3} = \{ (a,b)\mid a\)and\(b\) have same number of windows \(\} \)

Note thatIf 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 \({\rm{a}}\).

\({(a)_{{R_1}}} = \{ b\mid b\)has the same number of rooms as \(a\} \)

\({(a)_{{R_2}}} = \{ b\mid b\)has the same number of floors as \(a\} \)

\({(a)_{{R_3}}} = \{ b\mid b\)has the same number of windows as \(a\} \)