See below the relevant definitions that we need to know to solve this exercise.

Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.”

Definition: “A relation R on a set A is called reflexive if (a,a) ∈ R for every element a ∈ A.”

Definition: “A relation R on a set A is called symmetric if (b,a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A.”

Definition: “A relation R on a set A is called transitive if whenever (a,b)∈R and (b,c)∈R, then (a,c) ∈ R, for all a,b,c ∈ A.”

The definitions above are from the textbook “Discrete Mathematics and its applications” By Rosen.

There are many possible answers to this exercise. I’ll show you three possible answers and you can find more as a way of practicing your new skills.

## Table of Contents

- Students that have the same age
- Students that have the same gender
- Students that speak the same language

## Students that have the same age

R={(a,b)| a and b have the same age}

Reflexive: Every student has the same age as himself/herself. Therefore, it is reflexive.

Symmetric: If student a has the same age as student b, then student b has the same age as student a. Therefore, it is symmetric.

Transitive: If student a has the same age as student b, and student b has the same age as student c, it follows that student a has the same age as student c. Therefore, it is transitive.

This relation is an equivalence relation because it is reflexive, symmetric, and transitive.

An equivalence class consists of the set of students that have the same age.

[a]_{R}={b| b has the same age as a }

## Students that have the same gender

R={(a,b)| a and b have the same gender}

Reflexive: Every student has the same gender as himself/herself. Therefore, it is reflexive.

Symmetric: If student a has the same gender as student b, then student b has the same gender as student a. Therefore, it is symmetric.

Transitive: If student a has the same gender as student b, and student b has the same gender as student c, it follows that student a has the same gender as student c. Therefore, it is transitive.

This relation is an equivalence relation because it is reflexive, symmetric, and transitive.

An equivalence class in this case will consist of the set of students that have the same gender.

[a]_{R}={b| b has the same gender as a}

## Students that speak the same language

R={(a,b)| a and b speak the same language}

Reflexive: Every student speaks the same language as himself/herself. Therefore, it is reflexive.

Symmetric: If student a speak the same language as student b, then student b speaks the same language as student a. Therefore, it is symmetric.

Transitive: If student a speak the same language as student b, and student b speaks the same language as student c, it follows that student a speaks the same language as student c. Therefore, it is transitive.

This relation is an equivalence relation because it is reflexive, symmetric, and transitive.

An equivalence class in this case will consist of the set of students that speak the same language.

[a_{1}]_{R}={b| b speak Spanish}

[a_{2}]_{R}={b| b speak English}

[a_{3}]_{R}={b| b speak Chinese}

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