General Mathematics Flashcards - Part 1

Vocabulary flashcards generated from the General Mathematics First Part (2026-2027 Syllabus) tables, covering fundamental concepts of structures, vector spaces, topology in R^n, functions, sequences, series, limits, and continuity.

Subset

Given two sets $A$ and $B$, we say that $A$ is a subset of $B$ (in symbols $A \subseteq B$) if all elements of $A$ are also elements of $B$; that is, if $x \in A$ implies $x \in B$.

Intersection

Given two sets $A$ and $B$, their intersection $A \cap B$ is the set of all elements that belong to both $A$ and $B$; that is, $x \in A \cap B$ if $x \in A$ and $x \in B$.

Empty set and disjoint sets

The empty set, denoted by $\emptyset$, is the set without elements. Two sets $A$ and $B$ are disjoint when they have an empty intersection, that is, $A \cap B = \emptyset$.

Union

Given two sets $A$ and $B$, their union $A \cup B$ is the set of all elements that belong to $A$ or to $B$; that is, $x \in A \cup B$ if $x \in A$ or $x \in B$.

Difference of sets

Given two sets $A$ and $B$, their difference $A - B$ is the set of all elements that belong to $A$, but not to $B$; that is, $x \in A - B$ if both $x \in A$ and $x \notin B$.

Complement of a set

Given a reference set or space $S$ and any of its subsets $A$, the difference $S - A$ is denoted by $A^c$ and is called the complement set of $A$.

De Morgan's laws

For any sets $A$ and $B$, $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.

Interval

A subset $I$ of $\mathbb{R}$ is an interval if, given any two points $x$ and $y$ of $I$ with $x \le y$, all points $z \in \mathbb{R}$ such that $x \le z \le y$ belong to $I$.

Upper and lower bounds

Let $A \subseteq \mathbb{R}$. A number $h \in \mathbb{R}$ is an upper bound of $A$ if $h \ge x$ for all $x \in A$, and a lower bound of $A$ if $h \le x$ for all $x \in A$.

Supremum and infimum

Given a set $A \subseteq \mathbb{R}$, the supremum of $A$ is its least upper bound ($\min(A^*)$), while the infimum is its greatest lower bound ($\max(A_*)$).

Extended real line

The set obtained by adding the two ideal points $+\infty$ and $-\infty$ to the real line, denoted by $\mathbb{R} \cup \{-\infty, +\infty\}$, $\bar{\mathbb{R}}$, or $[-\infty, +\infty]$.

Cartesian product of two sets

Given two sets $A_1$ and $A_2$, the Cartesian product $A_1 \times A_2$ is the set of all ordered pairs $(a_1, a_2)$ with $a_1 \in A_1$ and $a_2 \in A_2$.

Real vector

An element $x = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n$ is called a vector. The Cartesian product $\mathbb{R}^n$ is called the $n$-dimensional Euclidean space.

Inner product

Given vectors $x, y \in \mathbb{R}^n$, their inner product is the scalar $x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n = \sum_{i=1}^{n} x_i y_i$.

Weak, strict, and strong vector order

For $x, y \in \mathbb{R}^n$: weak order $x \ge y$ requires $x_i \ge y_i$ for all $i$; strict order $x > y$ requires $x_i \ge y_i$ for all $i$ with $x_i > y_i$ for at least one $i$; strong order $x \gg y$ requires $x_i > y_i$ for all $i$.

Euclidean norm

The norm of a vector $x \in \mathbb{R}^n$, denoted by $||x||$, is given by $||x|| = (x \cdot x)^{\frac{1}{2}} = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}$.

Cauchy–Schwarz inequality

Result stating that $|x \cdot y| \le ||x||\,||y||$ for all $x, y \in \mathbb{R}^n$, with equality holding if and only if the two vectors are collinear.

Orthogonal vectors

Two vectors $x, y \in \mathbb{R}^n$ are orthogonal (perpendicular), written $x \perp y$, if their inner product is zero ($x \cdot y = 0$).

Neighborhood or open ball

The set $B_{\varepsilon}(x_0) = \{x \in \mathbb{R}^n : d(x, x_0) < \varepsilon\}$ centered at $x_0 \in \mathbb{R}^n$ with radius $\varepsilon > 0$.

Interior point

A point $x_0 \in A$ for $A \subseteq \mathbb{R}^n$ is an interior point if there exists $\varepsilon > 0$ such that $B_{\varepsilon}(x_0) \subseteq A$.

Exterior point

A point $x_0 \in \mathbb{R}^n$ is exterior to $A$ if it is an interior point of the complement $A^c$, i.e., there exists $\varepsilon > 0$ such that $B_{\varepsilon}(x_0) \subseteq A^c$.

Boundary point

A point $x_0 \in \mathbb{R}^n$ is a boundary point of $A$ if it is neither interior nor exterior, meaning for every $\varepsilon > 0$, $B_{\varepsilon}(x_0) \cap A \neq \emptyset$ and $B_{\varepsilon}(x_0) \cap A^c \neq \emptyset$.

Limit or accumulation point

A point $x_0 \in \mathbb{R}^n$ is a limit point for $A$ if each neighborhood $B_{\varepsilon}(x_0)$ contains at least one point of $A$ distinct from $x_0$.

Derived set

The set of all limit points of a set $A$, denoted by $A'$.

Open set

A set $A \subseteq \mathbb{R}^n$ is called open if all its points are interior points, that is, if $\text{int}(A) = A$.

Closed set

A set $A \subseteq \mathbb{R}^n$ is called closed if it contains all its boundary points ($A = \bar{A}$), or equivalently if $A' \subseteq A$.

Compact set in $\mathbb{R}^n$

A set in $\mathbb{R}^n$ that is both closed and bounded.

Convex set

A set $C \subseteq \mathbb{R}^n$ is convex if, for every pair of points $x, y \in C$, the segment $\alpha x + (1 - \alpha) y \in C$ for all $\alpha \in [0, 1]$.

Function

Given sets $A$ and $B$, a function $f : A \rightarrow B$ is a rule that associates to each element of set $A$ exactly one element of set $B$.

Natural domain

For a function $f$, the largest set on which $f$ can be defined without becoming undefined or nonreal.

Cobb–Douglas function

A function $f : \mathbb{R}^n_+ \rightarrow \mathbb{R}$ defined by $f(x_1, \dots, x_n) = \prod_{i=1}^{n} x_i^{a_i}$ with exponents $a_i > 0$ such that $\sum_{i=1}^{n} a_i = 1$.

Level set and level curve

For a real-valued function $f : A \rightarrow \mathbb{R}$, the preimage $f^{-1}(k) = \{x \in A : f(x) = k\}$ for a given level $k \in \mathbb{R}$.

Injective function

A function $f : A \rightarrow B$ is injective (or one-to-one) if $x \neq y \implies f(x) \neq f(y)$ for all $x, y \in A$.

Surjective function

A function $f : A \rightarrow B$ is surjective (or onto) if $\text{Im}(f) = B$.

Bijective function

A function $f : A \rightarrow B$ that is both injective and surjective.

Concave and convex functions

A function $f : C \rightarrow \mathbb{R}$ on a convex set $C$ is concave if $f(\alpha x + (1 - \alpha) y) \ge \alpha f(x) + (1 - \alpha) f(y)$ for all $x, y \in C$ and $\alpha \in [0, 1]$, and convex if $f(\alpha x + (1 - \alpha) y) \le \alpha f(x) + (1 - \alpha) f(y)$.

Convergence of a real sequence

A sequence $\{x_n\}$ converges to $L \in \mathbb{R}$ ($x_n \rightarrow L$) if for every $\varepsilon > 0$ there exists $n_{\varepsilon} \ge 1$ such that $n \ge n_{\varepsilon} \implies |x_n - L| < \varepsilon$.

Little-o for sequences

For sequences $\{x_n\}$ and $\{y_n\}$ with $x_n$ eventually non-zero, $y_n = o(x_n)$ if $\frac{y_n}{x_n} \rightarrow 0$ as $n \rightarrow \infty$.

Asymptotic equivalence of sequences

For sequences $\{x_n\}$ and $\{y_n\}$ with $x_n$ eventually non-zero, $y_n \sim x_n$ if $\frac{y_n}{x_n} \rightarrow 1$ as $n \rightarrow \infty$.

Convergent series

A series $\sum_{n=1}^{\infty} x_n$ is convergent with sum $S$ if its sequence of partial sums $s_n = \sum_{i=1}^{n} x_i$ satisfies $\lim_{n \rightarrow \infty} s_n = S \in \mathbb{R}$.

Absolute convergence

A series $\sum_{n=1}^{\infty} x_n$ is absolutely convergent if the series of its absolute values $\sum_{n=1}^{\infty} |x_n|$ is convergent.

Continuity at a point

A function $f : A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$ is continuous at a limit point $x_0 \in A$ if $\lim_{x \rightarrow x_0} f(x) = f(x_0)$.

Weierstrass theorem

Theorem stating that every continuous function $f$ defined on a nonempty compact set $C$ attains both a maximum and a minimum on $C$.

Bolzano's zero-value theorem

Theorem stating that if $f : [a, b] \rightarrow \mathbb{R}$ is continuous and $f(a) f(b) \le 0$, then there exists $c \in [a, b]$ such that $f(c) = 0$.

Coercive function

A function $f : A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$ is coercive on $C \subseteq A$ if there is a scalar $t$ such that the upper contour set {x \in C : f(x) \ge t} is non-empty and compact.

Tonelli's theorem

Theorem stating that if $f$ is continuous and coercive on $C$, then there exists $x^* \in C$ such that $f(x^*) = \max_{x \in C} f(x)$.

Supercoercive function

A function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is supercoercive if $||x_n|| \rightarrow +\infty \implies f(x_n) \rightarrow -\infty$ for every sequence $\{x_n\}$ in $\mathbb{R}^n$.