Answer to Question #188070 in Discrete Mathematics for kenneth

Question

Let A = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4}

 and define a relation R on A as follows:

For all x, y  A, x R y ⇔ 3|(x − y).

It is a fact that R is an equivalence relation on A. Use set-roster notation to write the equivalence classes of R.

[0]=  

[1]=  

[2]=  

[3]=  

How many distinct equivalence classes does R have?

 

List the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets.)

  

Answer 1

Solution:

A = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4}

[0] contains all elements of A that are multiples of 3.

So, [0]={-3,0,3}

[1] contains all elements of A that leave a remainder of 1 when divided by 3.

So, [1]={-4,-1,1,4}

[2] contains all elements of A that leave a remainder of 2 when divided by 3.

So, [2]={-5,-2,2,}

[3]=[0].

So there are 3 distinct equivalence classes.

These are [0], [1], [2].