Modular arithmetic and equivalence classes

Question

I completely understand the concept of modulus arithmetic and grasp all the main ideas. The only thing I don’t understand is equivalence classes.

The formal definition is:

Another interpretation is that modular arithmetic deals with all the integers, but divide them into N equivalence classes, each of the form $\{i + kN \mid k \in\mathbb{Z}\}$ for some $i$ between $0$ and $N-1$.

Could someone explain this in layman’s terms with an example possibly? I have absolutely no idea what equivalence classes are.

Answer 1

The notion of an equivalence class always arises when dealing with something called an "equivalence relation." So to explain the former, we'll first discuss the latter.

An equivalence relation is (speaking broadly) a generalized notion of equality. For example, if we are conducting a survey, and we are only concerned with gender, we might say that two people are "equivalent" if they have the same gender. Similarly, in the case of modular arithmetic, we say that two numbers are equivalent if they have the same remainder when divided by some number $n$. So, if we are working mod $4$, we say that $1 \equiv 5$. That does not mean that $1=5$, just that they are equivalent for our purposes.

More formally, an equivalence relationship needs to be reflexive (everything should be equivalent to itself), symmetric (if $A$ is equivalent to $B$, $B$ is equivalent to $A$) and transitive (if $A$ is equivalent to $B$, and $B$ is equivalent to $C$, $A$ is equivalent to $C$). Thus, this generalized notion of equality behaves very similarly to the traditional "$=$".

When we construct an equivalence relationship, one can think of it as putting on a pair of coarse glasses, so that we can no longer distinguish between "equivalent" objects. Thus, mod $4$, it appears to us that $1,5,9,\cdots$ are all the same thing. This entire family of objects that are equivalent is called an equivalence class. When we are working mod $4$, there will be $4$ equivalence classes, one corresponding to all things with remainder $0$, another for all things with remainder $1$, etc. Moreover, when you add two things with remainder $2$, you get something with remainder $0$. Thus, we can say that when working mod $4$, there are only four objects: families of "equivalent" numbers, and we do arithmetic with these families.

Answer 2

Given $N$, we will make $N$ "boxes", labeled $0$, $1$, $2,\ldots,N-1$. We will divide all the integers among the boxes as follows: given an integer $a$, divide $a$ by $N$ with remainder; put $a$ in the box corresponding to its remainder.

So, for example, if $N=11$, then we have boxes labeled $0,1,\ldots,10$; in the box labeled $0$ we have all multiples of $11$; in the box labeled $1$ we have all numbers of the form $11k + 1$, with $k\in\mathbb{Z}$; and so on: $$\begin{align*} {}[0] &= \{\ldots, -22, -11, 0, 11, 22, 33,\ldots\} = \{0+11k\mid k\in\mathbb{Z}\};\\ {}[1] &= \{\ldots, -21, -10, 1, 12, 23, 34, \ldots\} = \{1+11k\mid k\in\mathbb{Z}\};\\ {}[2] &= \{\ldots, -20, -9, 2, 13, 24, 35,\ldots\} = \{2+11k\mid k\in\mathbb{Z}\};\\ &\vdots\\ {}[10] &= \{\ldots,-12, -1, 10, 21, 32, 43,\ldots\} = \{10+11k\mid k\in\mathbb{Z}\}. \end{align*}$$

Each of this boxes is an "equivalence class": we consider two integers to be "equivalent [modulo $N$]" if they are in the same box. Note that every integer is in some box, and no integer is in more than one box; the boxes partition the integers.

We can then define addition of boxes as follows: to add the box $[a]$ with the box $[b]$, take any number $x\in[a]$, any number $y\in[b]$, add them like you add integers normally, and then find the one and only one box $[c]$ such that $x+y\in[c]$. We define $[a]+[b]$ to be the box $[c]$.

This definition requires that you show that it is "well-defined": the definition of $[a]+[b]$ seems to depend on the numbers $x$ and $y$ that are chosen; we need to show that if you pick a different $x'\in[a]$ and a different $y'\in[b]$, then $x'+y'$ will be in the same box as $x+y$. This can be done directly, or as a consequence of structure that these boxes have and is derived from the general algebraic theory of "congruences".

In a sense, we are dealing with all integers now, because the boxes include all the integers. We can then define a relation on the integers by saying that "$a$ is congruent to $b$ modulo $N$" if and only if $a$ and $b$ are in the same box. We write this $a\equiv b\pmod{N}$. Then what I discussed above says that if $x\equiv x'\pmod{N}$ and $y\equiv y'\pmod{N}$, then $x+y\equiv x'+y'\pmod{N}$.

You can define modular arithmetic as regular arithmetic in terms of the boxes.

Answer 3

In very layman's terminology, you can think of an analogy of students in a class(room).

The three conditions for an Equivalence relation can be imagined as such:

Reflexive: Every student is in the same class as himself.

Symmetric: If student A is in the same class as student B, then student B is in the same class as student A.

Transitive: If student A is in the same class as student B, and student B is in the same class as student C, then student A is in the same class as student C.

In the context of modular arithmetic (e.g. modulo 6), one can see that the three conditions hold.

Then the integers in a certain class all have the the same reminder (modulo 6).

So for example, 1,7,13 are all in the same class, denoted by [1], as they all have remainder 1 when divided by 6.

Hope this analogy helps.