0.1: Sets

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Sets

Sets are collections of objects,
Sets are normally denoted by capital letters.
Elements are denoted by lowercase letters.
Elements: Let $A$ be a set. Then

$x\in A$	$x$ is an element of $A$
$x\notin A$	$x$ is not an element of $A$

How do we test if an element belongs to a set?

Example $\PageIndex{1}$

$M_{22}(\mathbb{R}):=$ set of all 2 x 2 matrices with real numbers, where $\mathbb{R}:=$ set of all real numbers.

Let $A=\big{\{}\begin{bmatrix}x\end{bmatrix} \in M_{22}(\mathbb{R}) \Bigg|\begin{bmatrix}x\end{bmatrix}\begin{bmatrix}1 &1\\
0 &1\end{bmatrix}=\begin{bmatrix}1 &1\\
0 &1\end{bmatrix}\begin{bmatrix}x\end{bmatrix}\big{\}}$.

Determine whether the following entries belong to $A$.

$\begin{bmatrix}1 & 0\\
0 & 0\end{bmatrix}$ does not belong to $A$.

Answer

$\begin{bmatrix}1 & 0\\
0 & 0\end{bmatrix} \cdot \begin{bmatrix}1 &1\\
0 &1\end{bmatrix}=\begin{bmatrix}1 &1\\
0 &0\end{bmatrix}$ and

$\begin{bmatrix}1 &1\\
0 &1\end{bmatrix} \cdot \begin{bmatrix}1 & 0\\
0 & 0\end{bmatrix}=\begin{bmatrix}1 &0\\
0 &0\end{bmatrix}$

Since $\begin{bmatrix}1 &1\\
0 &0\end{bmatrix} \ne \begin{bmatrix}1 &0\\
0 &0\end{bmatrix}$, $\begin{bmatrix}1 & 0\\
0 & 0\end{bmatrix} \notin A$.◻

$\begin{bmatrix}0 & 1\\
0 & 0\end{bmatrix}$ does belong to $A$.

Answer

$\begin{bmatrix}0 & 1\\
0 & 0\end{bmatrix} \cdot \begin{bmatrix}1 &1\\
0 &1\end{bmatrix}=\begin{bmatrix}0 &1\\
0 &0\end{bmatrix}$

$\begin{bmatrix}1 &1\\
0 &1\end{bmatrix} \cdot \begin{bmatrix}0 & 1\\
0 & 0\end{bmatrix}=\begin{bmatrix}0 &1\\
0 &0\end{bmatrix}$

Since the matrices are equal, $\begin{bmatrix}0 & 1\\
0 & 0\end{bmatrix} \in A$.◻

$\begin{bmatrix}0 & 0\\
1 & 0\end{bmatrix}$ does not belong to $A$.

Answer

$\begin{bmatrix}0 & 0\\
1 & 0\end{bmatrix} \cdot \begin{bmatrix}1 &1\\
0 &1\end{bmatrix}=\begin{bmatrix}0 &0\\
1 &1\end{bmatrix}$

$\begin{bmatrix}1 &1\\
0 &1\end{bmatrix} \cdot \begin{bmatrix}0 & 0\\
1 & 0\end{bmatrix}=\begin{bmatrix}1 &0\\
1 &0\end{bmatrix}$

Since $\begin{bmatrix}0 &0\\
1 &1\end{bmatrix} \ne \begin{bmatrix}1 &0\\
1 &0\end{bmatrix}$, $\begin{bmatrix}0 & 0\\
1 & 0\end{bmatrix} \notin A$.◻

$\begin{bmatrix}0 & 0\\
0 & 1\end{bmatrix}$ does not belong to $A$.

Answer

$\begin{bmatrix}0 & 0\\
0 & 1\end{bmatrix} \cdot \begin{bmatrix}1 &1\\
0 &1\end{bmatrix}=\begin{bmatrix}0 &0\\
0 &1\end{bmatrix}$

$\begin{bmatrix}1 &1\\
0 &1\end{bmatrix} \cdot \begin{bmatrix}0 & 0\\
0 & 1\end{bmatrix}=\begin{bmatrix}0 &1\\
0 &1\end{bmatrix}$

Since $\begin{bmatrix}0 &0\\
0 &1\end{bmatrix} \ne \begin{bmatrix}0 &1\\
0 &1\end{bmatrix}$, $\begin{bmatrix}0 & 0\\
0 & 1\end{bmatrix} \notin A$.◻

$\begin{bmatrix}0 & 0\\
0 & 0\end{bmatrix}$. does belong to $A$.

Solution

This one is obviously true, thus $\begin{bmatrix}0 & 0\\
0 & 0\end{bmatrix} \in A$.

Notations:

$\cup$ or (union)

$
A\cup B
=
\{x\mid x\in A\text{ or }x\in B\}.
$

$\cap$ and (intersection)

$
A\cap B
=
\{x\mid x\in A\text{ and }x\in B\}.
$

$\{\}^c$ complement of a set (aka $\{\}^{'}$).

$
A^c
=
\{x\mid x \notin A\}.
$

$A \backslash B$ is equivalent to $A \cap B^{'}$.

$
A \backslash B
=
\{x\mid x\in A\text{ and }x \notin B\}.
$

$\mathbb{Z}=$ set of all integers $=\{\ldots,-2,-1,0,1,2,\ldots\}$. A countable set.
$\mathbb{Q}=$ set of all rational numbers $=\{\dfrac{a}{b}: a,b \in \mathbb{Z}, b \ne 0 \}$. A countable set.
$\mathbb{Q}^c=$ set of all irrational numbers $=\{\pm e, \pm \pi, \pm \sqrt{2}, \ldots\}$. Not a countable set.
$\mathbb{R}=$ set of all real numbers $=\mathbb{Q} \cup \mathbb{Q}^c$. Note $\mathbb{Q} \cap \mathbb{Q}^c=\emptyset$. Not a countable set.
$\mathbb{C}=$ set of all complex numbers.
$M_{mn}(\mathbb{R}):=$ set of all $m \times n$ matrices over real numbers.
$M_{mn}(\mathbb{C}):=$ set of all $m \times n$ matrices over complex numbers.
The cardinality of a set $A$ is written as $|A|$. If $|A|$ is finite, then $A$ is called a finite set; otherwise, it is called an infinite set.

Example $\PageIndex{2}$

Let $A=\{1,2,3,5\}$. Therefore $|A|=4$. In this case, $A$ is called a finite set.

Subsets: $Screen Shot 2023-06-27 at 12.52.27 PM.png$

- Every element of B is an element of A, written as $ B\subseteq A.$
- Trivial subsets of $A$ are $A$ and $\emptyset $ (empty set).

Example $\PageIndex{3}$

$\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}$.

$Screenshot 2026-09-10 at 4.41.25 PM.png$

Example $\PageIndex{4}$

Let $A=\{1,2,3,5\}$. Therefore $|A|=4$. In this case, $A$ is called a finite set.

Definition: Power set

Let $A$ be a set. Then the power set of $A$,denoted $P(A), =$ the set of all subsets of $A=\{ B \mid B\subseteq A\}.$

\(x\in A\)	\(x\) is an element of \(A\)
\(x\notin A\)	\(x\) is not an element of \(A\)

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Notations:

Example \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Definition: Power set