What is $ \mathbb{N}^2 / R $?

Question

I understand, that R is a symbol for an equivalence relation and $\mathbb{N}$ are the natural numbers.

But what is meant with $\mathbb{N}^2 / R $ ?

Answer 1

This is the set of all equivalence classes of $R$.

Answer 2

An equivalence relation $R$ on a set $S$ tells us whether two elements $a,b$ are related or not.

• in simple language, $a$ and $b$ are friends

So when considering an integer $a$, we can define $\bar a=\{x\in S\mid xRa\}$ of all elements that are in relation with $a$.

• $\bar a=friends(a)$

Note that when $a$ and $b$ are in relation then $\bar a=\bar b$

• the friend of my friend is my friend too

In the end, we can split $S$ into classes of related elements.

• $S$ is split into clans, all people are friends into one clan

The quotient $S/R$ is the set of all equivalence classes of $R$

• we are not interested in people anymore, just $clan_1,clan_2,clan_3,...$

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Let's take an example in $\mathbb N$

We define $a\,R\,b\iff a\text{ and }b\text{ have same parity}$

We see there are $2$ classes, the even numbers and the odd numbers.

They can be represented by $\bar 0=\{\text{even numbers}\}$ and $\bar 1=\{\text{odd numbers}\}$, but we could have chosen as well $\bar 2$ and $\bar 7$, or $\bar 8$ and $\bar 1$, it does not matter which representative we choose, they are the same classes of even and odd numbers.

Now $\mathbb N/R=\{\bar 0,\bar 1\}=\{\text{even numbers},\text{odd numbers}\}$

Note that $\mathbb N/R$ has only $2$ elements (classes), but their nature is different from integers, they are themselves sets of integers.

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Let's take another example in $\mathbb N$

We define $a\,R\,b\iff |a-b| \text{ is divisible by 7 }$

This time there are $7$ classes, for sake of simplicity we will note them $\bar 0,\bar 1,...,\bar 6$

Now $\mathbb N/R$ is the set of all classes of numbers modulo $7$

$\bar r=\{x\in\mathbb N\mid x\equiv r\pmod 7\}$

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In your question, you asked for $\mathbb N^2/R$, so there is some relation between points in the plane with positive integer coordinates.

$\mathbb N^2/R$ is the set of classes of points that share the same property.

• For instance, such a relation $R$ could be: "the points belong to the same vertical line"

This means that we do not care about the second coordinate, thus $(n,a)$ and $(n,b)$ belong to the same class.

We could as well consider only the classes $(n,\square)$ or simply $\bar{n}$.

In this case, $\mathbb N^2/R$ can be identified to $\mathbb N$.

• For instance, such a relation $R$ could be: "the points belong to the same line issued from origin"

These lines all have a positive slope, which since it passes through points $(a,b)$ with integer coordinates, is a rational number.

In this case, $\mathbb N^2/R$ can be identified to $\mathbb Q^+$.

Answer 3

A very important equivalence relation on $\mathbb{N}^2$ is $$(a,b)\simeq (c,d) \iff a+d = b+c$$

( one thinks of it in terms of $a-b = c-d$, but the difference might not lie in $\mathbb{N}$, so we stay inside $\mathbb{N}$.

The equivalence classes can be indentified with $\mathbb{Z}$.