Archangel Macsika
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.)
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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].