Mathematics and Statistics Lecture Notes

Flashcards covering mathematical concepts from set theory and linear algebra to calculus and statistics.

Insieme

A non-ordered collection of non-repeated objects, which can be finite or infinite.

Elementi

Objects that belong to a set, usually represented by lowercase letters such as $x$, $y$, or $z$.

Insieme vuoto

The unique set that contains no elements.

Sottoinsieme

A smaller set contained within another set; for example, $B$ is a subset of $A$ when all elements of $B$ are also in $A$.

Unione

The set containing all elements of $A$ and $B$ without repetition.

Intersezione

The set containing only the common elements belonging to both $A$ and $B$.

Differenza tra due insiemi

The set containing only the objects of $A$ that are not in $B$, and vice versa.

Insieme dei numeri naturali ($N$)

The set containing only positive integers: $1, 2, 3, 4,\dots$

Insieme dei numeri interi relativi ($Z$)

The set of integers including positive numbers, negative numbers, and zero.

Insieme dei numeri razionali ($Q$)

All numbers that can be written as a fraction between two integers, where the denominator is non-zero.

Insieme dei numeri reali ($R$)

The set including both rational and irrational numbers, such as $\frac{1}{2}$, $\sqrt{2}$, and $\pi$.

Minimo

The smallest number within an set.

Massimo

The largest number within a set.

Minorante

A real number that is smaller than or equal to all elements of a set, even if it is not inside the set.

Maggiorante

A real number that is larger than or equal to all elements of a set, even if it is not inside the set.

Estremo inferiore (infimo)

The largest of all the lower bounds (minoranti) of a set.

Estremo superiore (supremo)

The smallest of all the upper bounds (maggioranti) of a set.

Insieme limitato

A set that has both a lower limit and an upper limit.

Sistema lineare quadrato

A set of equations where unknowns appear only to the first degree and the number of equations equals the number of unknowns.

Matrice dei coefficienti

A table of numbers consisting of the coefficients placed before the unknowns in a linear system.

Determinante

A numerical value associated with a square matrix; if it is $\neq 0$, there is a unique solution, and if it is $= 0$, there are infinite or no solutions.

Regola di Sarrus

A method for calculating the determinant of $3 \times 3$ square matrices by summing products of descending diagonals and subtracting products of ascending diagonals.

Piano Cartesiano

A two-dimensional coordinate system formed by two perpendicular axes: abscissas ($x$) and ordinates ($y$).

Punto medio

The point exactly halfway between two endpoints of a segment, calculated by the average of their coordinates: $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

Coefficiente angolare ($m$)

A value indicating the slope of a line, given by the tangent of the angle the line forms with the $x$-axis: $m = \tan(\theta)$.

Rette parallele

Lines that possess the same slope ($m$).

Rette perpendicolari

Lines where the product of their slopes ($m_1 \times m_2$) is $-1$.

Coseno

The horizontal coordinate ($x$) of a point on the unit circle corresponding to a given angle.

Seno

The vertical coordinate ($y$) of a point on the unit circle corresponding to a given angle.

Potenza

A mathematical operation representing repeated multiplication of a base by itself as many times as indicated by the exponent.

Nèpero

The mathematician who introduced logarithms; the constant named after him is approximately equal to $e \approx 2.7$.

Logaritmo

The inverse operation of a power used to determine the exponent needed for a base to produce a specific number.

Logaritmo naturale

A logarithm where the base is the number of Nèpero ($e$).

Valore assoluto

The distance of a number from zero on the real number line.

Discriminante ($\Delta$)

The value derived from $b^2 - 4ac$ in a quadratic equation that determines the nature of the solutions.

Equazione fratta

An equation where the unknown appears in the denominator of a fraction.

Equazione irrazionale

An equation where the unknown appears under a radical sign.

Dominio

The set of real values for which a function is defined.

Immagine

The set of real values that a function assumes as the variable moves through the domain.

Funzione iniettiva

A function where different elements of the domain always have different images in the codomain.

Funzione suriettiva

A function whose image coincides exactly with its entire codomain.

Funzione biettiva

A function that is both injective and surjective simultaneously.

Teorema di Weierstrass

A theorem stating that if a function is continuous on a closed and limited interval, it must possess an absolute maximum and minimum.

Funzione monotona

A function that is either always increasing or always decreasing throughout its entire domain.

Punto di accumulazione

A point where every neighborhood contains infinitely many points of the domain, allowing a limit to be defined.

Teorema di unicità

A fundamental limit theorem stating that if a limit exists, it is unique.

Teorema di confronto

A theorem stating that if a function is trapped between two functions having the same limit, it must also share that same limit.

Teorema di permanenza del segno

A theorem stating that if a limit is positive or negative, the function maintains that sign in a neighborhood of the point.

Asintoto verticale

A line indicating that a function diverges as it approaches a specific point in the domain.

Asintoto orizzontale

A line describing a finite value that a function approaches as $x$ tends toward $\infty$ or $-\infty$.

Rapporto incrementale

A measure of how a function varies when the variable undergoes a small increment ($h$), defined as $\frac{f(x + h) - f(x)}{h}$.

Derivata

The limit of the incremental ratio as the increment $h$ approaches zero, representing the instantaneous slope of the function.

Punto angoloso

A point where a function is continuous but has a sudden change in direction, resulting in different finite left and right derivatives.

Cuspide

A point where a function is continuous but has a terminal sharp peak where derivatives tend toward $\pm \infty$.

Teorema di Fermat

States that if a continuous and differentiable function has a local maximum or minimum, its derivative at that point is zero.

Teorema de L'Hospital

A rule allowing the calculation of limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ by taking the limit of the ratio of the derivatives.

Primitiva

A function whose derivative returns the original starting function.

Integrale indefinito

The set of all possible primitives of a function, written with a constant $+C$.

Integrale definito

A number measuring the area between a function's curve and the $x$-axis over a specific interval.

Metodo di Cavalieri-Simpson

A numerical method for approximating a definite integral using parabolic interpolation between the endpoints and the midpoint.

Popolazione

The set of individuals or objects on which a statistical study is performed.

Campione

A subset of the population chosen for statistical analysis.

Unità statistica

Every single element belonging to a population.

Caratteri qualitativi

Categories that cannot be measured numerically, such as the presence/absence of defects.

Caratteri quantitativi

Properties that are measurable, such as pH, moisture, or microbial content.

Frequenza assoluta

The number of occurrences of a specific value within a data set.

Frequenza relativa

The ratio between the absolute frequency and the total number of observations.

Istogramma

A graph representing frequency distribution for quantitative characters using rectangles of proportional height.

Media

A value describing the central tendency of the data, calculated as the average.

Varianza

A measure of data dispersion relative to the mean.

Mediana

The value that divides an ordered set of data into two equal groups.

Quartili

Values that divide ordered data into four equal parts ($Q1$, $Q2$/Mediana, $Q3$).

IQR (Intervallo Interquartile)

A measure of central dispersion calculated as $Q3 - Q1$.

Boxplot

A graphical representation showing the distribution, median, and quartiles of a data set.

Esperimento aleatorio

An experiment whose outcome cannot be predicted with certainty, such as tossing a coin.

Evento

A subset of possible outcomes from a random experiment.

Distribuzione normale

A bell-shaped probability density curve that is symmetrical and describes values oscillating around a mean.

Funzione di ripartizione

A function indicating the accumulated probability up to a value $x$, tending from $0$ at $-\infty$ to $1$ at $+\infty$.