MathsGee Answers is Zero-Rated (You do not need data to access) on: Telkom | Dimension Data | Rain | MWEB
First time here? Checkout the FAQs!
x
MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB

0 like 0 dislike
11 views
What are equivalence relations in set theory?
in Mathematics by Bronze Status (8,688 points) | 11 views

1 Answer

0 like 0 dislike
Best answer

Definition: A binary relation on a set, $\mathrm{A}$, is a subset, $\mathrm{R}$, of $\mathrm{A} \times \mathrm{A} .$ Alternatively, we can say that a subset of $\mathrm{A} \times \mathrm{A}$ is a relation on $\mathrm{A}$.


Here is some notation


Notation: $\mathrm{a} \sim \mathrm{b}$ if $(a, b) \in R$. In other words, a is related to $\mathrm{b}$ if both $\mathrm{a}$ and $\mathrm{b}$ belong to $\mathrm{R}$.


Here is a simple example. Let's say that we have a set of students in a classroom. Then two students are related if they sit in the same row of seats.


Definition: A relation $\sim$ is called an equivalence relation if the following hold:

  1.  $\mathrm{a} \sim \mathrm{a} \forall a \in A$. This is called reflexivity.
  2. If $a \sim b \Rightarrow b \sim a$. This is called symmetry.
  3. If $\mathrm{a} \sim \mathrm{b}$ and $\mathrm{b} \sim \mathrm{c}, \Rightarrow \mathrm{a} \sim \mathrm{c}$. This is called transitivity.

and student $c$ are in the same row, then student a and student $c$ are in the same row. Therefore, it is an equivalence relation.

Lets look at another example.


Example:

Let $(a, b) \in \mathbb{Z}$. Show $\mathrm{a} \sim \mathrm{b}$ if $2 \mid(\mathrm{a}-\mathrm{b})$. That is, 2 divides $(\mathrm{a}-\mathrm{b})$

  1. If $\mathrm{a} \sim \mathrm{a}$, then $2 \mid(\mathrm{a}-\mathrm{a}) . \mathrm{So} \sim$ is reflexive.
  2. If $a \sim b \Rightarrow 2 \mid(a-b)$, then $2 \mid(b-a) \Rightarrow b \sim a .$ So $\sim$ is symmetric.
  3. If $a \sim b$ and $b \sim c, \Rightarrow 2 \mid(a-b)$ and $2|(b-c) . \Rightarrow 2|[(a-b)+(b-c)]$. That is, $2 \mid(a-c) . \Rightarrow a \sim c .$ So $\sim$ is transitive.

Therefore, $\sim$ is an equivalence relation.
Now we look at another important idea, the equivalence class.

by Bronze Status (8,688 points)

Related questions

1 like 0 dislike
1 answer
asked May 27 in Mathematics by Teddy Bronze Status (8,688 points) | 16 views
0 like 0 dislike
1 answer
asked Apr 25, 2020 in Mathematics by MathsGee Diamond (74,866 points) | 27 views
0 like 0 dislike
1 answer
asked Apr 25, 2020 in Mathematics by MathsGee Diamond (74,866 points) | 42 views
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true

MathsGee provides answers to subject-specific educational questions for improved outcomes.



On MathsGee Answers, you can:


1. Ask questions
2. Answer questions
3. Comment on Answers
4. Vote on Questions and Answers
5. Donate to your favourite users

MathsGee Tools

Math Worksheet Generator

Math Algebra Solver

Trigonometry Simulations

Vectors Simulations

Matrix Arithmetic Simulations

Matrix Transformations Simulations

Quadratic Equations Simulations

Probability & Statistics Simulations

PHET Simulations

Visual Statistics

ZeroEd Search Engine

Other Tools

MathsGee ZOOM | eBook