I need help with understanding this example on equivalence relation

Question

I know equivalence relation has to be reflexive , transitive and symmetric 

 

but this example has confused me

 

The relation R on A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
defined by
R = {(x, y) | x, y ∈ A and x, y have the same remainder
when divided by 3}

 

Is it reflexive? (1, 1), (2, 2), (3, 3), . . . , (12, 12) ∈ R X
Is it symmetric?
if (x, y) ∈ R this means x, y have the same remainder when
divided by 3,
then obviously y, x have the same remainder when divided by 3,
hence (y, x) ∈ R as well X
Is it transitive?
if (x, y) ∈ R this means x, y have the same remainder when
divided by 3,
if (y, z) ∈ R this means y, z have the same remainder when
divided by 3,
then also x, z have the same remainder when divided by 3,
hence (x, z) ∈ R X

I get the reflexive part because (1,1) 1/3 = 0.3333 ....(12,12)

 

but I do not get the rest can someone please give me example , is there a better way of finding out a equivalence relation

 

Thank you!

Answer 1

 

       I use the tilda symbol ~ to denote the relationship.

     The proof goes something like this:


REFLEXIVE:
Given x~x

then x and x have the same remainder when divided by 3.

so x = 3d + r for some integer d and integer remainder r.

x~x


SYMMETRIC:

Given x~y

then x and y have the same remainder when divided by 3.

then for some integer remainder r
x = 3*D + r for some integer D
y = 3* _d + r for some integer _d

So y and x have the same remainder when divided by 3.
y~x

TRANSITIVE:

Given: X~y and y~z
Then
x and y have the same remainder when divided by 3.
y and z have the same remainder when divided by 3.

Then for some integer remainder r:

x = 3*D + r for some integer D
y = 3d + r for some integer d , different from D
z = 3 _d + r for some integer _D

So x and z have the same remainder r when divided by 3
So x~z