Let $\sim $ be equivalence relation on $\mathbb N$ so there 4 different integers

Question

Let $\sim $ be equivalence relation on $\mathbb N$ so there are $4$ different integers: $i,j,k,m$ that for all $n\in \mathbb N$ there's $t\in \{i,j,k,m\}$ such that $n \sim t$ then:

A. $1\leq \left | \left \{ \left [ n \right ]:n\in \mathbb{N} \right \} \right |\leq 4$

B. $\left | \left \{ \left [ n \right ]:n\in \mathbb{N} \right \} \right |=4$

C. $\left | \left \{ \left [ n \right ]:n\in \mathbb{N} \right \} \right |>5$

D. $0\leq \left | \left \{ \left [ n \right ]:n\in \mathbb{N} \right \} \right |\leq 4$

Can someone explain what to do here and how to get to the answer.

thank you!

Answer 1

You are given an equivalence relation on $\Bbb N$, and you are told that there are four integers that any other integer is equivalent to at least one of those.

How many equivalence classes are there? I assume that you are supposed to mark the true statements. But for a correct answer on that you will have to ask your teachers.

HINT: Remember that if $a\sim b$ then $[a]=[b]$. This should get you a long way in the answers.

Answer 2

The question is asking for the cardinality of the set of equivalence classes, where $n_1 \sim n_2$ if and only if $[n_1] = [n_2]$.

What can you conclude in response to the question: how many classes must exist if for every natural number is related to exactly one of $\{i, j, k, m\}$? Use the definition of an equivalence relation, how this equivalence relation is defined, and the fact that an equivalence relation partitions the set on which it is defined.

Note also we are given that there exist $4$ different (unique) integers $i, j, k, m$ such that *every natural number $n$ is equivalent to (exactly) one of $i, j, k, m$. So there are at least $4$ equivalence classes, and at most four equivalence classes under $\sim$. Why?