What does this relation represents ?

Question

so i have a hard time understanding what would this relation looks like, we aren't given any precise function so it's hard to know what this would look like. We have to establish the relation and then say if it's an equivalence relation and give the differents equivalence classes. My representation of this relation would be : ${ <0,0> , <1,1>, <2,2>, <3,3> }$ but im not quite sure if i'm right. Then i would say it is an equivalence relation but we cannot define the different classes since we aren't given any function. Here's the function and the relation R that we have to create ( N represents natural numbers) :

f: N$\,\to\,${0,1,2,3} a function. Construct the relation R included in $N^2$ the relation ( has the same image by f) which means :

Thank you for your help !!

Answer 1

Your answer makes no sense, because $R$ is an equivalence relation in $\mathbb N$, not in $\{0,1,2,3\}$.

You should check that $R$ is an equivalence relation (that's easy). Then there are (at most) $4$ equivalence classes:

• $\{n\in\mathbb{N}\,|\,f(n)=0\}$;
• $\{n\in\mathbb{N}\,|\,f(n)=1\}$;
• $\{n\in\mathbb{N}\,|\,f(n)=2\}$;
• $\{n\in\mathbb{N}\,|\,f(n)=3\}$.

Of course, each of these sets is really an equivalence class when it is not empty. In particular, there are $4$ equivalence classes if and only if $f$ is surjective.

Answer 2

It concerns a relation on $\mathbb N$ (so not on $\{0,1,2,3\}$ as you seem to think).

The equivalence classes form a partition of set $\mathbb N$ and the class that is represented by element $n\in\mathbb N$ is the set:$$[n]=\{m\in\mathbb N\mid f(m)=f(n)\}$$

The number of equivalence classes corresponds with the cardinality of the image of $f$.

This because the classes take the shape: $$\{n\in\mathbb N\mid f(n)=i\}$$ where $i$ ranges over the image of $f$.

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More generally any equivalence relation $R$ on any set $X$ can be presented like this.

If $P$ denotes the set of equivalence classes of $R$ then we can prescribe function $\nu:X\to P$ by $x\mapsto[x]$ where $[x]$ denotes the equivalence class that is represented by $x$.

Then automatically we have:$$\forall,y\in X\; [xRy\iff \nu(x)=\nu(y)]$$