Mathematics

Problem

Let

A=\{1,2,3,4,5,6,7,8,9\}

be partitioned into the subsets

B_1=\{1,4,7\},\quad B_2=\{2,5,8\},\quad B_3=\{3,6,9\}.

Using this partition, respond to the following: a. (i) Define the equivalence relation

R

A

that is induced by the given partition. b. (ii) Prove that, for any

x,y\in A

x\,R\,y

if and only if

x\equiv y \pmod{3}

. c. (iii) Write the full equivalence class that contains the element

5

Intuitive explanation

This problem is about modular arithmetic and equivalence classes. Notice that $B_1$ , $B_2$ , and $B_3$ partition the set by grouping numbers that leave the same remainder when divided by 3—this is the key structure. Once you recognize that each subset contains all numbers congruent to the same value mod 3, you can use this pattern to solve whatever property or question follows about how elements relate within and across these groups.

Intuitive explanation

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