Set

A set is a collection of elements. Elements can be numbers, points, functions, etc.

Example

$A=\{2,4,6,8\}$
2, 4, 6, and 8 are all elements apart of the set $A$.
$2\in A$ means 2 is an element of $A$ or 2 is in $A$.

Special sets:

$\mathbb{N}=\{1,2,3,4,5,6,7,...\}$
$\mathbb{Z}=\{...,-3,-2,-1,0,1,2,3,4,...\}$
$\mathbb{R}=AllRealNumbers$
$\mathbb{Q}=AllRationalNumbers$

Sets can have other sets as elements. Consider $E=\{1,\{2,3\},\{2,4\}\}$, which has three elements: the number 1, the set $\{2,3\}$ and the set $\{2,4\}$. Thus $1\in E$ and $\{2,3\}\in E$ and $\{2,4\}\in E$. But note that $2\notin E$, $3\notin E$ and $4\notin E$.

If $X$ is a finite set, then its cardinality or size is the number of elements in the set, denoted by $|X|$.

$\varnothing =\{\}$ is the empty set. $|\varnothing |=0$. The empty set is the only set with a cardinality of zero. $\{\varnothing \}\ne \varnothing$ . $|\{\varnothing \}|=1$ and $|\varnothing |=0$.

Set Builder Notation:

Used to describe sets that are too big or complex to list between braces.

Example

Consider the infinite set of even integers $E=\{...,-6,-4,-2,0,2,4,6,...\}$. In set builder notation: $E=\{2n:n\in \mathbb{Z}\}$

More generally:
$X=\{expression:rule\}$

Cartesian Product:

The Cartesian product of two sets $A$ and $B$ is another set, denoted as $A\times B$ and defined as $A\times B=\{(a,b):a\in A,b\in B\}$.

Example

if $A=\{k,l,m\}$ and $B=\{q,r\}$, then $A\times B=\{(k,q),(k,r),(l,q),(l,r),(m,q),(m,r)\}$.

Info

If A and B are finite sets, then $|A\times B|=|A|\cdot |B|$.

The set $\mathbb{R}\times \mathbb{R}=\{(x,y):x,y\in \mathbb{R}\}$ is the set of points on a Cartesian plane.

Subsets:

Suppose $A$ and $B$ are sets. If every element of $A$ is also an element of $B$, then we say $A$ is a subset of $B$, and we denote this as $A\subseteq B$. We write $A⊈B$ if $A$ is not a subset of $B$, that is, if it is not true that every element of $A$ is also an element of $B$. Thus $A⊈B$ means that there
is at least one element of $A$ that is not an element of $B$.

The empty set is a subset of all sets, that is, $\varnothing \subseteq B$ for any set $B$.

If a finite set has $n$ elements, then it has $2^n$ subsets.

Power Sets:

If $A$ is a set, the power set of $A$ is another set, denoted as $\mathcal{P}(A)$ and defined to be the set of all subsets of $A$. In symbols, $\mathcal{P}(A)=\{X:X\subseteq A\}$.

Example

Suppose $A=1,2,3$

$\mathcal{P}(A)=\{\varnothing ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$

If $A$ is a finite set, then $|\mathcal{P}(A)|=2^{|A|}$.

Union, Intersection, and Difference

The union of $A$ and $B$ is the set $A\cup B=\{x:x\in Aorx\in B\}$.
The intersection of $A$ and $B$ is the set $A\cap B=\{x:x\in Aandx\in B\}$.
The difference of $A$ and $B$ is the set $A-B=\{x:x\in Aandx\notin B\}$.

Example

Suppose $A=\{a,b,c,d,e\}$, $B=\{d,e,f\}$ and $C=\{1,2,3\}$.

1. $A\cup B=\{a,b,c,d,e,f\}$
2. $A\cap B=\{d,e\}$
3. $A-B=\{a,b,c\}$

Complement

Let $A$ be a set with a universal set $U$. Universal sets: $\mathbb{N},\mathbb{Z},\mathbb{R}$. The complement of $A$, denoted $\bar{A}$, is the set $A=U-A$.

Multiset

A multiset is a hybrid of a set and a list; in a multiset, elements can be
repeated, but order does not matter.

The number of k-element multisets that can be made from the
elements of an $n$-element set $X=\{x_1,x_2,...,x_n\}$ is
$\left(\begin{array}{c}k+n-1\\ k\end{array}\right)=\left(\begin{array}{c}k+n-1\\ n-1\end{array}\right)$