1.2 Sets

Definition

A set is a collection of elements.

We denote sets by capital letters like $X$ and $Y$. If an element $x$ belongs to the collection $X$, we write $x\in X$. If $y$ does not belong to $X$, we write $y\notin X$. There is a special set containing no elements. This set is called an empty set and is denoted by $\varnothing$.

If you think of a set you should imagine a sack full of elements. The sack is your set and the elements in the sack are the elements belonging to the set. An empty set becomes a sack with no elements inside.

Constructors

To define a set we need to specify the elements inside the set. For doing that, we use the following notation

$X=\left\{X\right|$ condition on $x\}$

Here, we mean that the set $X$ consists of all elements $x$ such that the condition on $x$ holds.

e.g. The set of natural numbers

$\mathbb{N}=\left\{x\right|x$ is a natural number$\}=\{0,1,2,3,\cdots ,n,\cdots \}$

Operations on sets

Intersection

If we are given two sets $X$ and $Y$, then we define the intersection of $X$ and $Y$ as follows

$X\cap Y=\left\{z\right|z\in X\wedge z\in Y\}$

If we denote the sets $X$ and $Y$ by discs on a plain then the intersection of $X$ and $Y$ is denoted as below

Union

If we are given two sets $X$ and $Y$, then we define the union of $X$ and $Y$ as follows

$X\cup Y=\left\{z\right|z\in X\vee z\in Y\}$

If we denote the sets $X$ and $Y$ by discs on a plain then the union of $X$ and $Y$ is denoted as below

Difference

If we are given two sets $X$ and $Y$, then we define the difference between $X$ and $Y$ as follows

$X\setminus Y=\left\{z\right|z\in X\wedge z\notin Y\}$

If we denote the sets $X$ and $Y$ by discs on a plain then the difference between $X$ and $Y$ is denoted as below

Cartesian product

If we are given two sets $X$ and $Y$, then their Cartesian product is defined as follows

$X\times Y=\left\{(x,y)\right|x\in X;y\in Y\}$

The Cartesian product is simply the set of all possible pairs $(x,y)$ where the first element is taken from $X$ and the second from $Y$.