{%hackmd @RintarouTW/About %}

# Set Theory

## 1.1 Set Notation and Relations

Set : A collections of elements.
Set **A**, if $x\in A$, $x$ is an element of **A**. Or $x\notin A$, $x$ is not an element of **A**.
$$
\cases{
A=\{1,2,3\}\\
\mathbb{N}:the\ natrual\ numbers, \{1, 2, 3, \ldots\}\\
\mathbb{Z}:the\ integers, \{\ldots,-3, -2, -1, 0, 1, 2, 3, \ldots\}\ as\ a\ set\\
\mathbb{Q}:the\ rational\ numbers\ as\ a\ set\\
\mathbb{R}:the\ real\ numbers\ as\ a\ set\\
\mathbb{C}:the\ complex\ numbers\ as\ a\ set\\
\mathbb{P}:the\ positive\ integers\ as\ a\ set\\
}
$$

Set-Builder Notation

$$
\mathbb{Q} = \{a/b\mid a,b\in\mathbb{Z},b\neq 0\}\\
\mathbb{C} = \{a+bi\mid a,b\in\mathbb{R}, i^2=-1\}\\
$$

Finite Set: the set has a finite number of elements.

Cardinality: the number of the elements in the set, denoted |A|.

Subset: $\forall a\in A\implies a\in B$ , denoted $A\subseteq B$,
ex: $\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}\subseteq{C}$

Set Equality: $(A\subseteq B) \land (B\subseteq A)$, denoted $A=B$

:::info
we don't care the order of the elements in the sets.
:::

Empty set/Null set: $\emptyset$

Improper subset: for any set A, A is called an improper subset of A.

Proper subset: for other subsets of A(including $\emptyset$) are called proper subset of A.

## 1.2 Basic Set Operations

Intersection: $A\cap B = \{x\mid x\in A\ and\ x\in B\}$

Solving the system of linear equations could be viewed as an intersection.
$$
\cases{
x+y=7\\
x-y=3
}\implies \cases{
A=\{(x,y)\mid x+y=7\}\\
B=\{(x,y)\mid x-y=3\}\\
}\\
A\cap B = (5,2)
$$

Disjoint Sets: $A\cap B =\emptyset$

Union: $A\cup B=\{x\mid x\in A\ or\ x\in B\}$

Universe: The universe, or universal set, is the set of all elements under discussion for possible membership in a set. We normally reserve the letter *U* for a universe in general discussions.

### 1.2.2 Set Operations and their Venn Diagrams


Complement of a set: $B-A = \{x\mid x\in B\ and\ x\notin A\}$

If *U* is the universal set, then $U-A$ is denoted by $A^c$ and is called the complement of A. $A^c = \{x\mid x\in U\ and\ x\notin A\}$

Symmetric Difference: (Exclusive) $A\oplus B = (A\cup B) - (A\cap B)$

## 1.3 Cartesian Products and Power Sets

Cartesian Product: A,B are sets. $A\times B = \{(a,b) \mid a\in A\ and\ b\in B\}$, (a,b) is an ordered pair.

ex: Cartesian coordinate system could be viewed as Cartesian products of the same infinite set($\mathbb{R}$).

$|A\times B| = |A| \times |B|$

$A^n = \overbrace{A\times A\times\cdots\times A}^{n}$

### 1.3.2 Power Sets

Power Set: If A is any set, the power set of A is the set of all subsets of A, denoted $\mathcal{P}(A)$.

ex:
- $\mathcal{P}(\emptyset)=\emptyset$
- $\mathcal{P}(\{1\})=\{\emptyset,\{1\}\}$
- $\mathcal{P}(\{1,2\})=\{\emptyset,\{1\}\,\{2\}\,\{1,2\}\}$

## 1.4 Binary Representation of Positive Integers

#### Grouping by Twos

## 1.5 Summation Notation and Generalizations

$$
\cases{
\sum_\limits{k=1}^n a_k = a_1 + a_2 +\cdots+a_n\\
\bigcap_\limits{k=1}^n a_k = a_1 \cap a_2 \cap\cdots\cap a_n\\
\bigcup_\limits{k=1}^n a_k = a_1 \cup a_2 \cup\cdots\cup a_n\\
\bigoplus_\limits{k=1}^n a_k = a_1 \oplus a_2 \oplus\cdots\oplus a_n\\
\prod_\limits{k=1}^n a_k = a_1 \cdot a_2 \cdot\cdots\cdot a_n\\
\mathop{\vcenter{\huge\times}}_\limits{k=1}^n{a_k}= (a_1,a_2,\ldots,a_n)
}
$$

###### tags: `math` `set`