How to deal with equivalence relations and equivalence classes

Question

I have the following relation $m^3=n^3$ on $\mathbb{Z}$.

I know how to show that it is an equivalence relation but I am facing a problem in finding the equivalence classes can u help me please?

Answer 1

The point of a relation is that you have a set $X$ of things and you declare that two of them are "related" or "the same" if some sort of property holds between them.

An equivalence relation is one that satisfies some extremely nice properties (ones that you would expect to want to use) such as:

Reflexivity: "anything is related to itself", i.e. $a\sim a$ for any $a\in X$.

Symmetry: "relatedness is irrelevant of order", i.e. if $a\sim b$ then $b\sim a$.

Transitivity: "relations can be glued", i.e. if $a\sim b$ and $b\sim c$ then $a\sim c$.

For example the notion of "$=$" is always an equivalence relation. In some sense equivalence relations are generalizations of this (where we don't necessary want to measure being the exact same thing but some weaker property).

Another example that isn't equality is the notion of being congruent mod $n$. Here $X = \mathbb{Z}$ and we say $a\sim b$ if $n\mid (a-b)$. We write this as $a\equiv b \bmod n$.

Now given an equivalence relation we can play the following game. Write down an element and find all things related to it. This is like the "family" of the element (all of its relations). We call this the equivalence class of the element.

It turns out that for equivalence relations a nice thing happens. Everything in $X$ lies in a unique equivalence class, i.e. the equivalence classes partition $X$ into pieces.

For example there are two equivalence classes on $X=\mathbb{Z}$ under the mod $2$ equivalence relation, the sets consisting of odd numbers and even numbers.

The equivalence classes of $=$ are always of the form $\{x\}$ since only $x$ can equal $x$.

So on to your question. I see a relation but no set to put it on! This really does matter (do you have any chairs in your family?).

If we assume the set is $\mathbb{Z}$ with the relation $m\sim n$ if $m^3 = n^3$ then clearly the equivalence classes are just singleton sets $\{m\}$ (since $m^3 = n^3$ implies $m=n$ in $\mathbb{Z}$).

However if instead we place this relation on $\mathbb{C}$ then the equivalence classes are of the form $\{z, \zeta_3 z, \zeta_3^2 z\}$ for $z\neq 0$ and $\{0\}$ (since now taking cube roots gives three answers unless $z=0$).

So you see to fully answer your question I need to know a set! You could be placing this relation on numbers, matrices, polynomials, ... and all would give different equivalence classes.