Sets: Definitions and Cardinality

Generalities About Sets

• A set refers to a collection of objects.
• Cardinality refers to the number of elements in a set denoted as Card(E).
• A void set includes no elements and denoted as $\emptyset$, its cardinality is Card($\emptyset$) = 0.
• A singleton is a set containing only one element.
• With inclusion $A \subset B$, every element of A is an element of B.
• Equality means $A = B$ if $A \subset B$ and $B \subset A$.
• Union ($A \cup B$) represents elements that belong to A or B.
• Intersection ($A \cap B$) represents elements that belong to both A and B.
• Difference ($A \setminus B$) represents elements that belong to A but not to B.
• Disjoint sets means sets A and B are disjoint if $A \cap B = \emptyset$.
• A partition of a set E is an ensemble of non-void subsets of E, two at a time disjointed, whose juncture is equivalent to E.

Cardinal of Sets

• Card($A \cup B$) = Card($A$) + Card($B$) - Card($A \cap B$)
• Card($A \cup B \cup C$) = Card($A$) + Card($B$) + Card($C$) - Card($A \cap B$) - Card($A \cap C$) - Card($B \cap C$) + Card($A \cap B \cap C$)
• Card($A \setminus B$) = Card($A$) - Card($A \cap B$)
• When ($A \subset E$), Card($\bar{A}$) = Card($E$) - Card($A$) where $\bar{A}$ represents the complement of A in E.

P-tuples

• Elements of E refers to a ordered series of p elements of E.
• Represented as: $(x_1, x_2,..., x_p)$ with $x_i \in E$ for any i.

Arrangements

• The number of arrangements of p elements among n (p $\le$ n) refers to the number of p-tuples of distinct elements of E.
• $A_n^p = n(n-1)(n-2)...(n-p+1) = \frac{n!}{(n-p)!}$

Permutations

• Number of permutations of n elements refers to the number of arrangements of n elements among n.
• $A_n^n = n!$

Combinations

• Number of combinations of p elements among n (p $\le$ n) refers to the number of subsets of E containing p elements (order doesn't count).
• $C_n^p = \binom{n}{p} = \frac{n!}{p!(n-p)!}$

Properties of Binomial Coefficients

• $\binom{n}{p} = \binom{n}{n-p}$
• $\binom{n}{p} + \binom{n}{p+1} = \binom{n+1}{p+1}$ (Pascal's Formula)
• $\sum_{k=0}^{n} \binom{n}{k} = 2^n$
• $\sum_{k=0}^{n} \binom{n}{k} x^k = (1+x)^n$ (Newton's Binomial)

Multiensembles

• Refers to a collection of objects in which the same object can appear various times.
• The number of means to choose p elements among n with authorized repetitions: $\binom{n+p-1}{p}$

Summary Table

Definition

• $\mathbb{C} = {a + bi \mid a, b \in \mathbb{R}}$ where $i^2 = -1$

Operations

• Let $z_1 = a + bi$ and $z_2 = c + di$.
• Addition: $z_1 + z_2 = (a + c) + (b + d)i$
• Multiplication: $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$

Geometric Representation

• A complex number $z = a + bi$ can be represented by a point $(a, b)$ in the complex plane.

Polar Form

• $z = r(\cos \theta + i \sin \theta)$
  • $r = |z| = \sqrt{a^2 + b^2}$ is the modulus of $z$
  • $\theta = \arg(z)$ is the argument of $z$

Euler's Formula

• $e^{i\theta} = \cos \theta + i \sin \theta$

Moivre's Theorem

• $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$

Definition of Vector Spaces

• A vector space over a field $\mathbb{K}$ (often $\mathbb{R}$ or $\mathbb{C}$) is a set $V$ with two operations:
  • Addition: $+: V \times V \rightarrow V$
  • Scalar multiplication: $\cdot: \mathbb{K} \times V \rightarrow V$
• Vector space operations must satisfy a series of axioms, including associativity and commutativity of addition, existence of neutral and inverse elements for addition, compatibility of scalar multiplication, existence of a neutral element for scalar multiplication, and distributivity.

Vector Space Examples

• $\mathbb{R}^n$ is a vector space over $\mathbb{R}$.
• $\mathbb{C}^n$ is a vector space over $\mathbb{C}$.
• The set of polynomials with coefficients in $\mathbb{K}$ is a vector space over $\mathbb{K}$.
• The set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a vector space over $\mathbb{R}$.
• $m \times n$ matrices over $\mathbb{R}$.

Vector Subspaces

• A subset $W$ of a vector space $V$ is a subspace if $W$ is itself a vector space with the same operations as $V$.
• $W$ of $V$ is a vector subspace if and only if:
  • $W$ is non-empty
  • $u + v \in W$ for all $u, v \in W$
  • $a \cdot u \in W$ for all $a \in \mathbb{K}$ and $u \in W$

Linear Combinations

A linear combination of vectors $v_1, v_2,..., v_n$ is a vector of the form:

• $a_1v_1 + a_2v_2 +... + a_nv_n$ where $a_1, a_2,..., a_n \in \mathbb{K}$.

Linear Span

• The linear span of vectors $v_1, v_2,..., v_n$, denoted $\text{span}(v_1, v_2,..., v_n)$, is the set of all linear combinations of $v_1, v_2,..., v_n$.

Linear Independence

• Vectors $v_1, v_2,..., v_n$ are linearly independent if the unique solution to the equation $a_1v_1 + a_2v_2 +... + a_nv_n = 0$ is $a_1 = a_2 =... = a_n = 0$.
• Otherwise, they are linearly dependent.

Basis

• A basis of a vector space $V$ is a set of linearly independent vectors that span $V$.

Dimension

• The dimension of a vector space $V$ is the number of vectors in a basis of $V$.

Rank of Matrix

• The rank of a matrix is the dimension of the vector space spanned by its columns.

Concept of Real Variable Vector Function

• A real variable vector function is a function $\overrightarrow{r}: \mathbb{R} \rightarrow \mathbb{R}^n$ that assigned each existing real quantity t a vector $\overrightarrow{r}(t)$ of n composition.
• If described mathematically: $$\begin{aligned} \vec{r}: \mathbb{R} & \longrightarrow \mathbb{R}^{n} \ t & \longrightarrow \vec{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right) \end{aligned}$$
• $f_i(t)$ are real functions of a real variable, referred to as the component functions of $\overrightarrow{r}(t)$.

Vector Function Graphics

• The graph of a vector $\overrightarrow{r}(t)$ in $\mathbb{R}^2$ or $\mathbb{R}^3$ is the dot compilation $(\mathrm{x}, \mathrm{y})=(\mathrm{f}(\mathrm{t}), \mathrm{g}(\mathrm{t}))$ or $(\mathrm{x}, \mathrm{y}, \mathrm{z})=(\mathrm{f}(\mathrm{t}), \mathrm{g}(\mathrm{t}), \mathrm{h}(\mathrm{t}))$, respectively, for any t quantities in the $\overrightarrow{r}$ domain.
• In other words, the graph of $\overrightarrow{r}(t)$ is the curve featured by the end of the vector $\overrightarrow{r}(t)$ when t varies.
• For example, the vector ($\overrightarrow{r}(t)=(\cos (t), \operatorname{sen}(t))$) can be graphed as a round with a single radio centered.

Domain of one vector function

• The domain of a vector function $\overrightarrow{r}(t)$ is the quantity of all t values for all component functions $f_i(t)$, which are defined as: $$\operatorname{Dom}(\overrightarrow{\mathrm{r}})=\operatorname{Dom}\left(\mathrm{f}{1}\right) \cap \operatorname{Dom}\left(\mathrm{f}{2}\right) \cap \ldots \cap \operatorname{Dom}\left(\mathrm{f}_{\mathrm{n}}\right)$$
• For example:
  • If $\overrightarrow{\mathrm{r}}(\mathrm{t})=\left(\sqrt{\mathrm{t}+1}, \frac{1}{\mathrm{t}}\right)$ $$\begin{aligned} \operatorname{Dom}(\sqrt{\mathrm{t}+1}) & =[-1, \infty) \ \operatorname{Dom}\left(\frac{1}{\mathrm{t}}\right) & =\mathbb{R}-{0} \end{aligned}$$
  • Therefore, $\operatorname{Dom}(\overrightarrow{\mathrm{r}})=[-1,0) \cup(0, \infty)$.

Limit of a Vector Function

• The limit of a vector function $\overrightarrow{r}(t)$ when $t$ tends to $a$, is the vector $\overrightarrow{L}$ such that each of its components is the limit of the corresponding component of $\overrightarrow{r}(t)$.
• For example:
  • $\lim _{t \rightarrow a} \overrightarrow{\mathrm{r}}(\mathrm{t})=\left(\lim {\mathrm{t} \rightarrow \mathrm{a}} \mathrm{f}{1}(\mathrm{t}), \lim {\mathrm{t} \rightarrow \mathrm{a}} \mathrm{f}{2}(\mathrm{t}), \ldots, \lim {\mathrm{t} \rightarrow \mathrm{a}} \mathrm{f}{\mathrm{n}}(\mathrm{t})\right)$
  • Requires the limits of all component operations present.
• If $\overrightarrow{\mathrm{r}}(\mathrm{t})=\left(\mathrm{t}^{2}, \frac{\operatorname{sen}(\mathrm{t})}{\mathrm{t}}\right)$ $$\lim _{t \rightarrow 0} \vec{r}(t)=\left(\lim _{t \rightarrow 0} t^{2}, \lim _{t \rightarrow 0} \frac{\operatorname{sen}(t)}{t}\right)=(0,1)$$

Continuity of a Vector Function

• A Vector function $\overrightarrow{r}(t)$ is continuous in (t = a) if all the following conditions are present:
  • $\overrightarrow{\mathrm{r}}(\mathrm{a})$ is defined.
  • $\lim _{t \rightarrow a} \overrightarrow{\mathrm{r}}(\mathrm{t})$ exists.
  • $\lim _{t \rightarrow a} \overrightarrow{\mathrm{r}}(\mathrm{t})=\overrightarrow{\mathrm{r}}(\mathrm{a})$
• In other words, a vector function is continuous at one point if single operating components at that point are continuous.

Matrix calculation in Finnish & English

Definition

• A matrix $A$ is a rectangular table of numbers
• $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$ or $A = (a_{ij}) \in \mathbb{R}^{m \times n}$
  • $i$ is row index
  • $j$ is column index

Square Matrix

• If $A$ is an $n \times n$ matrix, then $A$ is a square matrix.

Diagonal Matrix

• A diagonal matrix is a square matrix whose off-diagonal elements are zero.
• $A = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \ 0 & a_{22} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & a_{nn} \end{bmatrix}$

Identity Matrix

• The identity matrix $I$ is an $n \times n$ diagonal matrix whose diagonal elements are ones.
• $I = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix}$

Zero Matrix

• The zero matrix $O$ is an $m \times n$ matrix whose all elements are zeros.
• $O = \begin{bmatrix} 0 & 0 & \cdots & 0 \ 0 & 0 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 0 \end{bmatrix}$