The absolute value (modulus) of a real number \(a\) is a non-negative value \(\vert a \vert\) determined as follows \[ \vert a \vert = \left\{\eqalign{a \quad &\hbox{for} \quad a\geq 0 \cr -a \quad &\hbox{for} \quad a < 0 \cr} \right. \]

A directed angle is an ordered pair of rays starting at the same point. The first ray (looking at counter-clockwise) is called an initial side, and second one – a terminal side of the angle.

The cardinality of a finite set \(A\) is the number of its elements and is denoted with \(\overset{=}{A}\).

The circle with a centre at \(S\) and radius \(r\), where \(r>0\), is a set of points \(P\) in the plane, which are equidistant from a fixed point \(S\) and this distance is equal to \(r\), i.e. \[ \vert SP\vert =r, \] where \(\vert SP\vert\) means the length of the segment \(SP\).

The binomial coefficient is a number \({n \choose k}\) given by the formula \[ {n \choose k}={n!\over k!(n-k)!}, \] where \(n,k\in\mathbb{N}\) and \(k\leq n\).

A \(k\)-element combination of an \(n\)-element set (\(k\)-combination of \(n\)), where \(k\le n\), is each \(k\)-element subset of this set.

Let \(A\) be any set in the universe \(X\), i.e. \(A\subset X\). The complement of \(\class{km-czerwony}{A}\) in the space \(\class{km-niebieski}{X}\) is a set denoted as \(A'\), where \[ A' = \{x\in X:\ x\notin A\}=X\backslash A \]

A conjunction (logical product) of statements \(p\) and \(q\) is the statement \(p\:{ \wedge}\: q\), which is read: \(p\) and \(q\). The truth values of the conjunction are described in the truth table below:

\( p\)	\( q\)	\( p\;{ \wedge}\; q\)
1	1	1
1	0	0
0	1	0
0	0	0

Logical connectives (operators) are:

1. not (\(\sim \))
2. or (\(\vee \))
3. and (\(\wedge \))
4. implication (\(\Rightarrow \))
5. equivalence (\(\Longleftrightarrow\)).

Cosine of \(\alpha\) is equal to the length of the adjacent side to the angle \(\alpha\) divided by the length of the hypothenuse, i.e. \[ \cos \alpha ={{\class{km-niebieski}{b}}\over {\class{km-zielony}{c}}} \]

Cotangent of \(\alpha\) is equal to the length of the adjacent side to the angle \(\alpha\) divided by the length of the opposite side, i.e. \[ \cot \alpha ={{\class{km-niebieski}{b}}\over {\class{km-czerwony}{a}}} \]

The function \(f:\ D_f \longrightarrow \mathbb{R}\) is a bounded function on the set \(A\subset D_f\) if \[\exists_{m,M\in \mathbb{R}}\quad \forall_{x\in A}\quad m\leq f(x)\leq M\]

The function \(f:\ D_f \longrightarrow \mathbb{R}\) is a function bounded above on the set \(A\subset D_f\) if \[\exists_{M\in \mathbb{R}}\quad \forall_{x\in A}\quad f(x)\leq M\]

The function \(f:\ D_f \longrightarrow \mathbb{R}\) is a function bounded below on the set \(A\subset D_f\) if \[\exists_{m\in \mathbb{R}}\quad \forall_{x\in A}\quad f(x)\geq m\]

The function \(f:\ D_f \longrightarrow \mathbb{R}\) is a constant function on the set \(A\subset D_f\) if \[ \exists _{c\in \mathbb{R}}\quad\forall_{x\in A}\quad f(x)=c\]

The cosine function is a function \(\cos:\ \mathbb{R}\rightarrow \mathbb{R}\) given by \[ \cos x=\cos \alpha, \] where \(x\) is the angle measure \(\alpha\) in radians.

The cotangent function is a function \(\cot :\: \mathbb{R}\backslash \{k\pi: \: k\in \mathbb{Z}\} \rightarrow \mathbb{R}\) given by \[ \cot x=\cot \alpha, \] where \(x\) is the angle measure \(\alpha\) in radians.

The function \(f\) is a decreasing function on the set \(A\subset D_f\) if \[ \forall_{x_1,x_2\in A}\quad x_1<x_2\ \Longrightarrow\ f(x_1)>f(x_2)\]

The function \(f:\ D_f \longrightarrow Y\) and the function \(g:\ D_g\longrightarrow Y\) are equal if \[ D_f=D_g\quad\wedge\quad\forall_{x\in D_f}\ f(x)=g(x) \]

The function \(f:\ X\longrightarrow Y\) is an even function if \[\forall_{x\in X}\quad -x\in X \quad \wedge\quad f(-x)=f(x)\] The graph of an even function is symmetrical with resoect to the \(y\)-axis.

Let \(a\) be a real number such that \(a\in(0,1)\cup (1,\infty)\). The exponential function is a function \(f:\mathbb{R}\rightarrow \mathbb{R}_+\) given by the formula \[f(x)=a^x\]

The function \(f:\ D_f \longrightarrow \mathbb{R}\) is an increasing function on the set \(A\subset D_f\) if \[\forall_{x_1,x_2\in A}\quad x_1 <x_2\ \Longrightarrow\ f(x_1)<f(x_2)\]

The function \(f:\ D_f \longrightarrow \mathbb{R}\) is an injective function on the set \({A\subset D_f}\) if \[\forall_{x_1,x_2\in A}\quad x_1\not= x_2\ \Longrightarrow\ f(x_1)\not= f(x_2)\]

Let the function \(f:\ X \longrightarrow Y\) be a bijection. The inverse function of the function \(f\) is the function \(f^{-1}:\ Y\longrightarrow X\) which satisfies the following condition \[ \forall_{x\in X}\quad\forall_{y\in Y}\quad f^{-1}(y)=x\quad \Longleftrightarrow\quad y=f(x) \]

The linear function is a function \(f:\ \mathbb{R} \longrightarrow \mathbb{R}\) given by the formula \[f(x)=ax+b,\] where \(a,b\in \mathbb{R}\). The constant \(a\) is the gradient, and the constant \(b\) is the \(y\)-coordinate of the \(y\)-intercept of a linear function.

Let \(a\) be a real number such that \(a\in(0,1)\cup (1,\infty)\). The logarithmic function is a function \(f:\ \mathbb{R}_+\rightarrow \mathbb{R}\) given by the formula \[ f(x)=\log_a x \]

The function \(f\) is a monotonic function in the set \(A\subset D_f\) if it is either increasing, decreasing, non-increasing or non-decreasing on the set \(A\subset D_f\).

The function \(f\) is a non-decreasing function on the set \(A\subset D_f\) if \[\forall_{x_1,x_2\in A}\quad x_1<x_2\ \Longrightarrow\ f(x_1)\leq f(x_2)\]

The function \(f\) is a non-increasing function on the set \(A\subset D_f\) if \[\forall_{x_1,x_2\in A}\quad x_1<x_2\ \Longrightarrow\ f(x_1)\geq f(x_2)\]

The function \(f:X\longrightarrow Y\) is an odd function if \[\forall_{x\in X}\quad -x\in X \quad \wedge\quad f(-x)=-f(x)\] The graph of an odd function has rotational symmetry with respect to the origin \((0,0)\).

The function \(f:\ X\longrightarrow Y\) is a periodic function if \[\exists_{T>0}\quad \forall_{x\in X}\quad x+T\in X \quad \wedge\quad f(x+T)=f(x)\] The number \(T\) is called the period of the function \(f\). If there exists the smallest positive constant with this property, it is called the fundamental period.

The power function is a function \(f\) which can be shown by the formula \[f(x)=x^\alpha,\] where \(\alpha\) is a real number.

The quadratic function (quadratic trinomial) is a function \(f:\ \mathbb{R} \longrightarrow \mathbb{R}\) given by the formula \[f(x)=ax^2+bx+c,\] where \(a,b,c\in \mathbb{R}\) and \(a\neq 0\).

The rational function is a function \(f\) in the form of \[ f(x)={P(x)\over Q(x)},\] where \(P(x)\), \(Q(x)\) are polynomials, but the polynomial \(Q(x)\) is a non-zero polynomial.

The sine function is a function \(\sin:\ \mathbb{R}\rightarrow \mathbb{R}\) given by \[ \sin x=\sin \alpha, \] where \(x\) is the angle measure \(\alpha\) in radians.

The function \(f\) is a strictly monotone function on the set \(A\subset D_f\) if it is either increasing or decreasing on the set \(A\subset D_f\).

The tangent function is a function \({\tan:\: \mathbb{R}\backslash \{{\pi\over 2}+k\pi:\: k\in \mathbb{Z}\} \rightarrow \mathbb{R}}\) given by \[ \tan x=\tan \alpha, \] where \(x\) is the angle measure \(\alpha\) in radians.

The graph of the function \(f:\ X\longrightarrow Y\) is the set \[\left\{(x,y)\in \mathbb{R}^2:\ x\in X \ \wedge\ y=f(x)\right\}\]

An implication of statements \(p\) and \(q\) is the statement \(p\Rightarrow q\), which is read: \(p\) implies \(q\) or if \(p\) than \(q\). Statement \(p\) is the predecessor (hypothesis) of the implication and \(q\) is the successor (conclusion) of the implication. The truth values of the implication are described in the truth table below:

\( p\)	\( q\)	\( p\;\Rightarrow\; q\)
1	1	1
1	0	0
0	1	1
0	0	1

Let \(a\) be a real number such that \(a\in (0,1)\cup (1,\infty)\). A logarithm of a positive real number \(x\) to the base \(a\) is a number \(y\) such that \(a^y=x\), and is denoted as \(\log_ax\).This definition can be written symbolically \[\forall_{a\in \mathbb{R}_+\backslash\{1\}}\quad \log_ax=y\quad \Longleftrightarrow\quad a^y=x\] The logarithm \(\log_{10}x\) is called common logarithm and denoted as \(\log x\).

A magnitude of the vector \(\overrightarrow{u}=\left[u_1,u_2\right]\) \(\overrightarrow{u}=\left[u_1,u_2\right]\) equals \[ \vert \overrightarrow{u} \vert=\sqrt{\left(u_1\right)^2+\left(u_2\right)^2} \]

The expression ”for all \(x\)” is called a general (universal) quantifier and is denoted with the symbol \[\forall_{x}\quad\hbox{or}\quad\bigwedge_{x}\]

The polynomial \(Q(x)\) is the quotient, and the polynomial \(R(x)\) is the reminder of the division of the polynomial \(N(x)\) by the polynomial \(D(x)\) if for every \(x\in \mathbb{R}\) is satisfied by the condition \[ N(x) = D(x) \cdot Q(x) + R(x) \] and the degree of the remainder \(R(x)\) is smaller than the degree of the divisor \(D(x)\). If \(R(x)\equiv 0,\)then it is said that the polynomial \(N(x)\) is divided by the polynomial \(D(x)\). The above condition on the assumption that \(D(x)\neq 0\) can be written in the form \[ {N(x) \over D(x)} = Q(x) + {R(x) \over D(x)} \]

Let \(\alpha\) be a central angle with the radius \(r\). The radian of a subtended angle \(\alpha\) is the ratio of the arc length \(l\) to the radius of the circle, i.e. \[{\class{km-niebieski}{\alpha}} = {{\class{km-zielony}{l}}\over {\class{km-czerwony}{r}}}\] The unit of angle \(1\) is called a radian.

The number \(x_0\in \mathbb{R}\) is called a root of the polynomial \(P(x)\) if \[ P(x_0)=0 \]

Let \(k\) be a positive integer. The number \(x_0\) is a root of multiplicity \(k\) of the polynomial \(P(x)\) if the polynomial \(P(x)\) is divided by the polynomial \((x-x_0)^k\), and is not divided by the polynomial \((x-x_0)^{k+1}\).

The scalar multiplication is the operation of multyplying a vector (\overrightarrow{u}=\left[u_1,u_2\right]\) by the scalar \(\alpha\in\mathbb{R}\) which gives us the vector \[ \alpha\overrightarrow{u}=\left[\alpha u_1,\alpha u_2\right] \]

An arithmetic sequence is the sequence \(\left(a_n\right)\) if \[ \exists_{d\in \mathbb{R}}\quad \forall_{n\in \mathbb{N_+}}\quad a_{n+1}=a_n+d \] The number \(d\) is a common difference of an arithmetic sequence.

A bounded above sequence is the sequence \(\left(a_n\right)\) if \[\exists_{M\in \mathbb{R}}\quad \forall_{n\in \mathbb{N}_+}\quad a_n\leq M\]

A bounded below sequence is the sequence \(\left(a_n\right)\) if \[\exists_{m\in \mathbb{R}}\quad \forall_{n\in \mathbb{N}_+}\quad a_n\geq m\]

A bounded sequence is the sequence \(\left(a_n\right)\) if \[\exists_{m,M\in \mathbb{R}}\quad \forall_{n\in \mathbb{N}_+}\quad m\leq a_n\leq M\]

A constant sequence is the sequence \(\left(a_n\right)\)if \[\forall_{n\in \mathbb{N}_+}\quad a_n=a_{n+1} \] It means that any term of this sequence is the same number.

A decreasing sequence is the sequence \(\left(a_n\right)\) if \[\forall_{n\in \mathbb{N}_+}\quad a_n> a_{n+1}\] This condition occurs when every consecutive term is less than the preceding one.

A geometric sequence is the sequence \(\left(a_n\right)\) if \[ \exists_{q\in \mathbb{R}} \quad\forall_{n\in \mathbb{N}}\quad a_{n+1}=a_n\cdot q \] The number \(q\) is a common ratio of a geometric sequence.

An increasing sequence is the sequence \(\left(a_n\right)\) if \[\forall_{n\in \mathbb{N}_+}\quad a_n< a_{n+1}\] This condition occurs when every consecutive term is greater than the preceding one.

An indefinite sequence (or: sequence) is the function \[ f:\quad\mathbb{N}_+\longrightarrow \mathbb{R}\] The value of this function for any counting number \(n\) is an \(n\)th term of the sequence and is denoted by \(a_n\), i.e. \(f(n)=a_n\). The sequence is denoted by \(\left(a_n\right)\).

A non-decreasing sequence is the sequence \(\left(a_n\right)\) if \[\forall_{n\in \mathbb{N}_+}\quad a_n \leq a_{n+1}\] This condition occurs is not less than theeceding one.s="km-rysunek">

Tangent of \(\alpha\) is equal to the length of the opposite side to the angle \(\alpha\) divided by the length of the adjacent side, i.e. \[\tan \alpha ={{\class{km-czerwony}{a}}\over {\class{km-niebieski}{b}}} \]

A union of events \(A\) and \(B\) \((A\cup B)\) is a set of all outcomes from the sample space \(\Omega\), that favor the event \(A\) or \(B\).

A union of sets \(\class{km-czerwony}{A}\) and \(\class{km-zielony}{B}\) is a set denoted as \(A\cup B\), where \[ A\cup B= \{x:\ x\in A\ \vee\ x\in B\} \]

Let \(\overrightarrow{u}=\left[u_1,u_2\right]\) and \(\overrightarrow{v}=\left[v_1,v_2\right]\) be any vectors in \(\mathbb{R}^2\). It is said that the vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\) are equal vectors if \[ u_1=v_1 \quad \wedge \quad u_2=v_2, \] what can be written \(\overrightarrow{u}=\overrightarrow{v}\) in mathematical notation.

The vertex form of the quadratic function \(y=ax^2+bx+c\) is \[ y=a(x-p)^2+q, \] where \(p=-{b\over 2a}\) and \(q=-{\Delta\over 4a}\).

The zero of the function \(f\) is every argument \(x_0\in D_f\), for which \(f(x_0)=0\). Geometrically, the zero of the function \(f\) is an \(x\)-coordinate of the \(x\)-intercept.