Let $A$ and $B$ be sets.
$A\cup B$, the union of $A$ and $B$, is $\{x:x\in A\vee x\in B\}$.
$A\cap B$, the intersection of $A$ and $B$, is $\{x:x\in A\wedge x\in B\}$.
$A\setminus B$, the set difference of $A$ and $B$, is $\{x\in A:x\notin B\}$.
If $A$ is a subset of a set $\mathrm{\Omega}$ then ${A}^{c}$, the complement of $A$ in $\mathrm{\Omega}$, is $\{x\in \mathrm{\Omega}:x\notin A\}$.
We can express complements using set differences. If $A$ is a subset of $\mathrm{\Omega}$ then its complement ${A}^{c}$ in $\mathrm{\Omega}$ is equal to $\mathrm{\Omega}\setminus A$.
Suppose $A=\{0,1,2\},B=\{1,2,3\},C=\{4\}$.
$A\cup B=\{0,1,2,3\}$
$A\cap B=\{1,2\}$
$A\cap C=\mathrm{\varnothing}$
$A\setminus B=\{0\}$
$A\setminus C=A$
If $\mathrm{\Omega}=\{0,1,2,\mathrm{\dots}\}$ then ${A}^{c}$ would be $\{3,4,\mathrm{\dots}\}$.
The set $\mathbb{N}=\{0,1,2,\mathrm{\dots}\}$ is called the natural numbers. Some people exclude 0 from $\mathbb{N}$ but in MATH0005 the natural numbers include 0.
It’s typical to draw Venn diagrams to represent set operations. We draw a circle, or a blob, for each set. The elements of the set $A$ are represented by the area inside the circle labelled $A$. Here are some examples:
The size or cardinality of a set $X$, written $|X|$, is the number of distinct elements it has.
$|\{1,2\}|=2$
$|\mathrm{\varnothing}|=0$
$|\{1,2,1,3\}|=3$
A set is finite if it has 0, or 1, or 2, or any other natural number of elements. A set that is not finite is called infinite.
$\mathbb{N}$ and $\mathbb{Z}$ are infinite sets while the sets in Example 2.2.2 are all finite.