Thread: [SOLVED] Equivalance Relations

Can somebody clarify me how equivalence classes are obtained ?
And how can I answer this type of Qn ?

Qn:- Let S={1,2,3,4,5} have a partition consisting of the sets {1,3,5} and {2,4} .Show that this partition determines an equivalence relation .

An equivalence relation always forms a partition on the set in which it is defined. Each cell, or element, of the partition is an equivalence class.

For example, in your set:

{1,2,3,4,5} you have the partition
{1,3,5}
{2,4}

A good choice would be the equivalence relation:
element a of A is related to element b of A, if:
a is of the same parity as b
notice {1, 3, 5} are all odd,
{2,4} are both even.

I'll leave it up to you to confirm that this equivalence relation is in fact an equivalence relation:

It's reflexive, a R a
Symmetric, a R b implies b R a
Transitive.. a R b and b R c implies a R c....

a R b is the conventional notation used to denote element a is Related to b...

Also. Given an equivalence relation, an equivalence class is formally defined and denoted as the following:

Given an equivalence relation defined on some set A

[a] : {b ϵ A | b R a}

That is, the set of all the elements of A which are related to a.

As an example, consider the equivalence class of nonnegative integers congruent to 1 modulo 2:

[1(mod 2)] = {1, 3, 5, 7, 9, 11, etc} essentially, all the odd integers.

similarly the equivalence class of integers congruent to 0 mod 2:
[0 (mod 2)] = {0, 2, 4, 6, 8, 10, etch} all the even nonnegative integers.

The relation a R b if a is congruent to b modulo 2 is another equivalence relation that might help you with your question. The only equivalence classes of this relation are [0] and [1]. Together, these two equivalence classes partition all of the integers in the set on which this relation is defined.

But I wouldn't directly copy this. I'm a bit rusty on my notation.

Hello K A D C Dilshan

Originally Posted by K A D C Dilshan

Can somebody clarify me how equivalence classes are obtained ?
And how can I answer this type of Qn ?

Qn:- Let S={1,2,3,4,5} have a partition consisting of the sets {1,3,5} and {2,4} .Show that this partition determines an equivalence relation .

You might find it helpful to study the examples and my replies in this posting.

Grandad

Originally Posted by dr3amrunn3r

Also. Given an equivalence relation, an equivalence class is formally defined and denoted as the following:

Given an equivalence relation defined on some set A

[a] : {b ϵ A | b R a}

That is, the set of all the elements of A which are related to a.

As an example, consider the equivalence class of nonnegative integers congruent to 1 modulo 2:

[1(mod 2)] = {1, 3, 5, 7, 9, 11, etc} essentially, all the odd integers.

similarly the equivalence class of integers congruent to 0 mod 2:
[0 (mod 2)] = {0, 2, 4, 6, 8, 10, etch} all the even nonnegative integers.

The relation a R b if a is congruent to b modulo 2 is another equivalence relation that might help you with your question. The only equivalence classes of this relation are [0] and [1]. Together, these two equivalence classes partition all of the integers in the set on which this relation is defined.

Actually dr3amrunn3r I had this idea before .
Your definition of equivalence classes is not correct I think .
I think it should be corrected as follows
[a] = {x : (a,x) ∊ R} Isn`t it ?
so [1(mod2)] and [2(mod2)] are the partitions as I thought as we can not take 0 .
But I had no experience on building relations using partitions .
I think a is congruent to b modulo 2 is something similar to a≣b(mod2) i.e 2 | (a-b)
Thank you very much for confirming the solution .
If I have done any mistake please tell me .