Equivalence Classes

• Jul 21st 2011, 11:26 AM
lovesmath
Equivalence Classes
How do you find the distinct equivalence classes of R?

A = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}. R is defined on A as follows:
For all x, y elements of A, x R y <=> 3|(x-y).

I have the answer, but I do not understand the work leading up to the answer. I appreciate any help.

I am having the same issue with this one:
A = {-4, -3, -2, -1, 0, 1, 2, 3, 4}. R is defined on A as follows:
For all (m,n) elements of A, m R n <=> 5|(m^2 - n^2).
• Jul 21st 2011, 11:36 AM
Plato
Re: Equivalence Classes
Quote:

Originally Posted by lovesmath
How do you find the distinct equivalence classes of R?

A = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}. R is defined on A as follows:
For all x, y elements of A, x R y <=> 3|(x-y).

I have the answer, but I do not understand the work leading up to the answer. I appreciate any help.

I am having the same issue with this one:
A = {-4, -3, -2, -1, 0, 1, 2, 3, 4}. R is defined on A as follows:
For all (m,n) elements of A, m R n <=> 5|(m^2 - n^2).

In the first case you want \$\displaystyle x-y\$ to a multiple of 3.
So \$\displaystyle -4~\&~2\$ are in the same class because \$\displaystyle -4-2=-6\$ a multiple of 3.

You do the next one: multiples of 5.
• Jul 21st 2011, 11:58 AM
lovesmath
Re: Equivalence Classes
So for 3|(x - y), do you find the combinations of numbers that give you remainders 0, 1 and 2, and each of those combinations will be its own equivalence class?
• Jul 21st 2011, 12:05 PM
Plato
Re: Equivalence Classes
Quote:

Originally Posted by lovesmath
So for 3|(x - y), do you find the combinations of numbers that give you remainders 0, 1 and 2, and each of those combinations will be its own equivalence class?

All you do its look at each pair of numbers.
Is their difference a multiple of 3?
If yes, those two numbers belong to the same class.
If no, they are not in the same class.

So \$\displaystyle \{-4,-1,2,5\}\$ is one class.
• Jul 21st 2011, 02:08 PM
lovesmath
Re: Equivalence Classes
For the second relation, 5|(m^2 - n^2), I got the following equivalence classes:
{0}, {-4, -1, 1, 4}, {-3, -2, 2, 3}

Are these correct?
• Jul 21st 2011, 02:13 PM
Plato
Re: Equivalence Classes
Quote:

Originally Posted by lovesmath
For the second relation, 5|(m^2 - n^2), I got the following equivalence classes:
{0}, {-4, -1, 1, 4}, {-3, -2, 2, 3} Are these correct?

Yes they are correct. Good for you.