學海無涯，思境無維；數乃六藝，理之始也。
或有一得足矣 愚千慮

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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.

$$\{\begin{array}{l}A=\{1,2,3\}\\ \mathbb{N}:thenatrualnumbers,\{1,2,3,\dots \}\\ \mathbb{Z}:theintegers,\{\dots ,-3,-2,-1,0,1,2,3,\dots \}asaset\\ \mathbb{Q}:therationalnumbersasaset\\ \mathbb{R}:therealnumbersasaset\\ \mathbb{C}:thecomplexnumbersasaset\\ \mathbb{P}:thepositiveintegersasaset\end{array}$$

Set-Builder Notation

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

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

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

Subset:

$\mathrm{\forall }a\in A⟹a\in B$ , denoted

$A\subseteq B$,
ex:

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

Set Equality:

$(A\subseteq B)\wedge (B\subseteq A)$, denoted

$A=B$

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

Empty set/Null set:

$\mathrm{\varnothing }$

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

Proper subset: for other subsets of A(including

$\mathrm{\varnothing }$) are called proper subset of A.

1.2 Basic Set Operations

Intersection:

$A\cap B=\{x\mid x\in Aandx\in B\}$

Solving the system of linear equations could be viewed as an intersection.

$$\begin{gathered} \{\begin{array}{l}x+y=7\\ x-y=3\end{array}⟹\{\begin{array}{l}A=\{(x,y)\mid x+y=7\}\\ B=\{(x,y)\mid x-y=3\}\end{array} \\ A\cap B=(5,2) \end{gathered}$$

Disjoint Sets:

$A\cap B=\mathrm{\varnothing }$

Union:

$A\cup B=\{x\mid x\in Aorx\in B\}$

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:

$B-A=\{x\mid x\in Bandx\notin A\}$

If U is the universal set, then

$U-A$ is denoted by

$A^c$ and is called the complement of A.

$A^c=\{x\mid x\in Uandx\notin A\}$

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.

$A\times B=\{(a,b)\mid a\in Aandb\in B\}$, (a,b) is an ordered pair.

ex: Cartesian coordinate system could be viewed as Cartesian products of the same infinite set(

$\mathbb{R}$).

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

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

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:

• $\mathcal{P}(\mathrm{\varnothing })=\mathrm{\varnothing }$
• $\mathcal{P}(\{1\})=\{\mathrm{\varnothing },\{1\}\}$
• $\mathcal{P}(\{1,2\})=\{\mathrm{\varnothing },\{1\}\{2\}\{1,2\}\}$

1.4 Binary Representation of Positive Integers

Grouping by Twos

1.5 Summation Notation and Generalizations

$$\{\begin{array}{l}\sum _{k=1}^n{a}_k=a_1+a_2+\cdots +a_n\\ \bigcap _{k=1}^n{a}_k=a_1\cap a_2\cap \cdots \cap a_n\\ \bigcup _{k=1}^n{a}_k=a_1\cup a_2\cup \cdots \cup a_n\\ \underset{k=1}{\overset{n}{⨁}}{a}_k=a_1\oplus a_2\oplus \cdots \oplus a_n\\ \prod _{k=1}^n{a}_k=a_1\cdot a_2\cdot \cdots \cdot a_n\\ {\times }_{k=1}^n{a}_k=(a_1,a_2,\dots ,a_n)\end{array}$$

tags: math set