1. The problem statement, all variables and given/known data

Let R1 and R2 be the "congruent modulo 3" and "congruent modulo 4" relations on the set of integers.

2. Relevant equations

Find:

a) R1 U R2(or R1 union R2)

b)R1 intersects R2

There is also problem c, d but I won't write these here. If I am able to solve this, then the rest should be cake.

3. The attempt at a solution

I saw the answer but frankly I'm not quite sure how to solve it.

I know that a is congruent to b(mod 4) is the same as 4|a-b and that a is congruent to b(mod 3) is the same as 3|a-b

I checked the equivalence classes of R1 and R2

R1 has 3 being

[0]r1={...,-12,-9,-6,-3,0,3,6,9,12...}

[1]r1={...,-11,-8,-5-,-2,1,4,7,10,13,...}

[2]r2={...,-10,-7,-4,-1,2,5,8,11,14...}

R2 has 4

[0]r2={...,-12,-8,-4,0,4,8,12,..}

[1]r2={...,-11,-7,-3,1,5,9,13,...}

[2]r2={...,-10,-6,-2,2,6,10,...}

[3]r2={...,-9,-5,-1,3,7,11,...}

I have tried to express that the intersection of r1 and r2 is

{(a,b)|a is congruent to b(mod 12)} as the intersection of both r1 and r2 are values that can be divided by 12. However I'm not sure how to express this in a mathematical notation. Any help or guidance would be much appreciated!