# Thread: Set theory notation - help

1. ## Set theory notation - help

I am self studying graph theory - and a book has the notation:

The set $\displaystyle Z/nZ$ of intagers modulo n is written Zn (if you know how to use Latex to write the n as postscript characters pls explain - I have 4 PDF on latex and they all seem usless)

My question is - What doesthe set $\displaystyle Z/nZ$ mean - where $\displaystyle Z$ are integers?

2. Originally Posted by brennan
I am self studying graph theory - and a book has the notation:

The set Z/nZ of intagers modulo n is written Zn (if you know how to use Latex to write the integer sign and postscript characters pls explain - I have 4 PDF on latex and they all seem usless)

My question is - What doesthe set Z/nZ mean - where Z are integers?
$\displaystyle Z/3Z=\{[0],[1],[2]\}$
$\displaystyle Z/nZ=\{[0],[1],[2],\cdots,[n-1]\}$

3. To tell a fuller story, $\displaystyle n\mathbb{Z}=\{nx\mid x\in\mathbb{Z}\}$. It is an ideal in the ring $\displaystyle \mathbb{Z}$. Then $\displaystyle \mathbb{Z}/n\mathbb{Z}$ is a quotient ring.

There was time I was confused about the relationship between a quotient over a substructure (an ideal for rings and a normal subgroup for groups) and a quotient over a congruence (an equivalence relation that respects the structure's operations; in the case of rings, addition and multiplication). Well, the quotient ring $\displaystyle \mathbb{Z}/n\mathbb{Z}$ is $\displaystyle \mathbb{Z}/{\sim}$ where the congruence $\displaystyle \sim$ is defined as follows: $\displaystyle x\sim y$ if $\displaystyle x-y\in n\mathbb{Z}$. This means that $\displaystyle x\sim y$ iff $\displaystyle x\equiv y\pmod{n}$. And the quotient over a congruence is defined as the set of equivalence classes of the congruence. The properties of the congruence are used to make sure that operations on equivalence classes, which are defined by choosing representatives, are indeed well-defined.

However, one does not have to understand he above to understand $\displaystyle \mathbb{Z}_n$. The simplest way to think about it is: $\displaystyle \mathbb{Z}_n=\{0,\dots,n-1\}$ where addition and multiplication are ordinary addition and multiplication followed by taking a remainder when divided by n.

To produce $\displaystyle \mathbb{Z}_n$, type \mathbb{Z}_n.