# 0.1: Sets

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### Sets

• Sets are collections of objects,

• Sets are normally denoted by capital letters.

• Elements are denoted by lower-case letters.

• Subsets:

• Every element of B is an element of A, written as $$B\subset A.$$

• $$B=\{a \in A:$$ …..condition on okay!.

• Trivial subsets of $$A$$ are $$A$$ and $$\emptyset$$ (empty set).

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

Determine whether the following entries belong to $$A$$.

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

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

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

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

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

Since $$\begin{bmatrix}0 &1\\ 0 &0\end{bmatrix} = \begin{bmatrix}0 &1\\ 0 &0\end{bmatrix}$$, $$\begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix} \in A$$.◻

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

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

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

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

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

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

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

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

1. $$\begin{bmatrix}0 & 0\\ 0 & 0\end{bmatrix}$$.  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

• $$\cap$$ and / intersection

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

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

• $$\mathbb{Z}:=$$ set of all integers $$=\{\ldots,-2,-1,0,1,2,\ldots\}$$.  A countable set.

• $$\mathbb{Q}:=$$ set of all rational numbers $$=\{\frac{a}{b}: a,b \in \mathbb{Z}, b \ne 0 \}$$.   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 $$|A|$$.

•
##### Example $$\PageIndex{2}$$

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

Example $$\PageIndex{3}$$

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

This page titled 0.1: Sets is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.