Question

Let P1={B0, B1, B2} be a partition of Z, where B0={3n|n ∈ Z}, B1={3n+ 1|n ∈ Z}, and B2={3n+ 2|n ∈ Z}. Describe the equivalence relation R1 corresponding to P1.

Answer 1

By definition, $(a,b)\in R_1$ if and only if $a,b\in B_k$  for some $k\in\{1,2,3\}.$ The set $B_0=\{3n\ :\ n \in\mathbb Z\}$ contains all integers $a$  that have 0 as the remainder of the Euclidean division of $a$ by 3. The set $B_1=\{3n+1\ :\ n \in\mathbb Z\}$  contains all integers $a$ that have 1 as the remainder of the Euclidean division of $a$ by 3. And the set $B_2=\{3n+2\ :\ n \in\mathbb Z\}$ contains all integers $a$ that have 2 as the remainder of the Euclidean division of $a$ by 3.

   

Therefore, $(a,b)\in R_1$ if and only if $a$ and $b$ have the same remainder of the Euclidean division by 3.