Equivalent classes of sets (Is my solution correct)?

Question

Equivalence relation between two sets is defined as |S1|=|S2|. Use the equivalence defined above to partition the set {4, 5, 6, 34, 90, 87, 65, 21, 35, 22} into equivalence classes.

From what I understand we can have equivalence relations between two sets and the number of elements in the set can be from 1 to 11. This would mean there are equivalence classes such as [4]=[5]=...=[22]={4,5,6,...,35,22} and then [{4,5}]=[{5,6}] = ... ={members are sets of two elements}.

This would make the solution too big and uninteresting. Please correct me if I'm wrong.

The question is from peter linz's textbook "An Introduction to Formal Languages and Automata". The relation is defined in question 1.1.4 and the question in context is given in 1.1.14

Answer 1

The exact wording of the question (according to the fifth edition of the book):

Notice that this comes from the list of EXERCISES, not Examples

We are not wanting to use number $4$ above. As mentioned, this is not an equivalence relation on the set in exercise $14$. We are instead directed to EXAMPLE 1.4 below:

So, partition the set $\{2,4,5,6,9,23,24,25,31,37\}$ according to the relation $\equiv$ where $x\equiv y$ if and only if $x\mod{3}=y\mod{3}$

Group all of the multiples of three together, group all of the numbers which are one more than a multiple of three together, and group all of the numbers which are two more than a multiple of three together.

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As an aside, if you have an electronic version of the book, the blue highlighted phrase $\color{blue}{\text{Example 1.4}}$ is in fact a hyperlink which can be clicked taking you straight to the intended location.