Mathematics for Natural Sciences - Core Concepts and Terminology

Comprehensive vocabulary flashcards covering propositional logic, set theory, real and complex numbers, relations and functions, polynomials, transcendental functions, and analytic geometry of conic sections from the Mathematics for Natural Sciences lecture curriculum.

Proposition (or Statement)

A declarative sentence which has a truth value (either True or False, but not both).

Logical Conjunction

A compound proposition formed when two propositions $p$ and $q$ are joined with the connective "and" ($\wedge$); it is true if and only if both $p$ and $q$ are true.

Logical Disjunction

A compound proposition formed when two propositions $p$ and $q$ are joined with the connective "or" ($\vee$); it is false if and only if both $p$ and $q$ are false.

Logical Implication

A compound proposition formed when two propositions $p$ and $q$ are joined with the connective "implies" ($\Rightarrow$); it is false if and only if the hypothesis $p$ is true and the conclusion $q$ is false.

Logical Bi-implication (or Logical Equivalence)

A proposition formed when two propositions $p$ and $q$ are joined with the connective "bi-implication" ($\Leftrightarrow$); it is false if and only if $p$ and $q$ have different truth values.

Negation

A proposition formed by modifying a proposition $p$ to $\neg p$, whose truth value is $F$ if $p$ is $T$ and $T$ if $p$ is $F$.

Compound Statement

A proposition formed by joining two or more propositions using one or more logical connectives.

Tautology

A compound proposition that is always true regardless of the truth values of its component propositions.

Contradiction

A compound proposition that is always false regardless of the truth values of its component propositions.

Contingency

A compound proposition that is neither a tautology nor a contradiction.

Open Statement (or Predicate)

A sentence containing one or more variables whose truth value depends on the specific values assigned to those variables.

Universal Quantifier

The logical operator corresponding to the phrase "for every $x$" (or "for all $x$", "for each $x$"), symbolized by $\forall$.

Existential Quantifier

The logical operator corresponding to the phrase "there exists an $x$" (or "for some $x$", "for at least one $x$"), symbolized by $\exists$.

Counterexample

A specific individual element in the universe of discourse for which an open proposition $P(x)$ is false, demonstrating that $\forall x\, P(x)$ is false.

Argument (Logical Deduction)

An assertion that a given set of statements $p_1, p_2, \dots, p_m$, called hypotheses or premises, yields another statement $q$, called the conclusion.

Valid Argument Form

An argument form in which the conclusion $q$ is true whenever all the premises $p_1, p_2, \dots, p_m$ are simultaneously true.

Set

A well-defined collection of objects that share a certain property or properties, such that one can determine whether or not a given object belongs to the collection.

Empty (or Null) Set

The unique set that contains no elements, denoted by $\emptyset$ or $\{\}$.

Subset

A relation between sets where set $A$ is contained in set $B$ ($A \subseteq B$) if every element of $A$ is also an element of $B$.

Proper Subset

A relation between sets where set $A$ is a subset of set $B$ ($A \subset B$) such that every element of $A$ is in $B$, but $B$ has at least one element that is not in $A$.

Power Set

The set of all subsets of a given set $A$, denoted by $P(A)$; for a set with $n$ elements, it contains $2^n$ subsets.

Union of Sets

The set of all elements that belong to either set $A$ or set $B$ (or both), denoted by $A \cup B = \{x : x \in A \text{ or } x \in B\}$.

Intersection of Sets

The set of all elements that belong to both set $A$ and set $B$, denoted by $A \cap B = \{x : x \in A \text{ and } x \in B\}$.

Disjoint Sets

Two sets $A$ and $B$ that have no elements in common, meaning that $A \cap B = \emptyset$.

Relative Complement (Set Difference)

The set of all elements in $A$ that are not in $B$, denoted by $A \setminus B$ or $A - B$.

Absolute Complement

The set of all elements in the universal set $U$ that are not in $A$, denoted by $A'$ or $A^c$, equivalent to $U \setminus A$.

Symmetric Difference

The set of elements belonging to either set $A$ or set $B$ but not both, denoted by $A \Delta B = (A \setminus B) \cup (B \setminus A)$.

Venn Diagram

A pictorial representation of sets using interlocking regions (typically circles) enclosed within a rectangle that represents the universal set $U$.

Prime Number

A natural number greater than $1$ that has only two factors: $1$ and itself.

Composite Number

A natural number that has three or more factors.

Greatest Common Factor (GCF)

The greatest natural number that divides each of two or more given numbers without remainder.

Relatively Prime

A property of two numbers having a greatest common factor (GCF) equal to $1$.

Least Common Multiple (LCM)

The smallest natural number that is a multiple of each of two or more given numbers.

Well Ordering Principle

The mathematical axiom stating that every non-empty subset of the set of natural numbers contains a smallest (least) element.

Rational Number

Any number that can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

Irrational Number

A real number with a decimal expansion that is non-terminating and non-periodic, which cannot be expressed as a quotient of two integers.

Completeness Property of Real Numbers

The property stating that every non-empty subset of $\mathbb{R}$ that has an upper bound has a least upper bound (lub), and every non-empty subset that has a lower bound has a greatest lower bound (glb).

Complex Number

A number defined as an ordered pair $(x, y)$ of real numbers or written in the form $z = x + iy$, where $x, y \in \mathbb{R}$ and $i^2 = -1$.

Conjugate of a Complex Number

For a complex number $z = x + iy$, the number denoted by $\bar{z} = x - iy$, representing a reflection across the real axis.

Modulus (Norm) of a Complex Number

The non-negative real number $|z| = \sqrt{x^2 + y^2}$, which geometrically represents the distance from the point $(x, y)$ to the origin.

Principal Argument

The unique angle $\theta$ formed by a complex number $z$ with the positive real axis that satisfies the condition $-\pi < \theta \le \pi$, denoted by $\text{Arg}(z)$.

Polar Form of a Complex Number

The trigonometric representation $z = r(\cos(\theta) + i\sin(\theta))$, where $r = |z|$ is the modulus and $\theta$ is the argument.

De Moivre's Formula

The formula stating that for any complex number in polar form and any integer $n$, $[r(\cos(\theta) + i\sin(\theta))]^n = r^n(\cos(n\theta) + i\sin(n\theta))$.

Euler's Formula

The identity $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, allowing a complex number to be written in exponential form as $z = re^{i\theta}$.

Cartesian Product

The set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$, denoted by $A \times B = \{(a, b) : a \in A \text{ and } b \in B\}$.

Relation

Any subset of the Cartesian product $A \times B$ defined between a set $A$ and a set $B$.

Domain of a Relation

The set of all first coordinates of the ordered pairs in a relation $R$, denoted by $\text{Dom}(R) = \{a \in A : (a, b) \in R\}$.

Range of a Relation

The set of all second coordinates of the ordered pairs in a relation $R$, denoted by $\text{Range}(R) = \{b \in B : (a, b) \in R\}$.

Function

A relation in which each element of the domain corresponds to exactly one element of the range.

One-to-One (1-1) Function

A function $f : A \rightarrow B$ where no two distinct domain elements have the same image, meaning that $f(x_1) = f(x_2)$ implies $x_1 = x_2$.

Onto Function

A function $f : A \rightarrow B$ where every element in the codomain $B$ is the image of at least one element in the domain $A$, meaning $\text{Range}(f) = B$.

1-1 Correspondence

A function that is both one-to-one and onto.

Inverse Function

For a one-to-one function $f$, the relation $f^{-1} = \{(y, x) : (x, y) \in f\}$, which reverses the original mapping.

Composition of Functions

The operation combining two functions $f$ and $g$ defined by $(f \circ g)(x) = f(g(x))$ for all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Polynomial Function

A function defined by $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer, $a_i \in \mathbb{R}$, and $a_n \neq 0$.

Remainder Theorem

The algebraic theorem stating that when a polynomial $p(x)$ of degree at least $1$ is divided by $x - r$, the resulting remainder is $p(r)$.

Factor Theorem

The theorem stating that $x - r$ is a factor of a polynomial $p(x)$ if and only if $p(r) = 0$.

Fundamental Theorem of Algebra

The theorem stating that every polynomial $p(x)$ of degree $n \ge 1$ with complex coefficients has at least one zero in the complex number system.

Rational Function

A function expressed as the quotient of two polynomials, $f(x) = \frac{n(x)}{d(x)}$, where $d(x) \neq 0$.

Vertical Asymptote

The line $x = a$ where a function's values grow without bound ($f(x) \rightarrow \infty$ or $f(x) \rightarrow -\infty$) as $x$ approaches $a$ from the left or right.

Horizontal Asymptote

A horizontal line $y = b$ approached arbitrarily closely by the function values $f(x)$ as $x \rightarrow \infty$ or $x \rightarrow -\infty$.

Oblique (Slant) Asymptote

A non-horizontal line $y = ax + b$ ($a \neq 0$) approached by a rational function as $x \rightarrow \pm\infty$, occurring when the numerator's degree is exactly one higher than the denominator's degree.

Exponential Function

A function of the form $y = f(x) = b^x$, where the base $b > 0$ and $b \neq 1$, defined for all real values of $x$.

Logarithmic Function

The inverse of an exponential function, denoted by $y = \log_b(x)$, meaning $b^y = x$ for base $b > 0$ and $b \neq 1$.

Periodic Function

A function $f(x)$ for which there exists a positive number $p$ such that $f(x + p) = f(x)$ for all $x$ in its domain; the smallest such $p$ is called the period.

Amplitude (of a Periodic Function)

Half the difference between the maximum and minimum values of a periodic function: $A = \frac{1}{2}[\max(f(x)) - \min(f(x))]$.

Hyperbolic Sine Function

The function defined by combinations of exponential functions as $\sinh(x) = \frac{e^x - e^{-x}}{2}$.

Hyperbolic Cosine Function

The function defined by combinations of exponential functions as $\cosh(x) = \frac{e^x + e^{-x}}{2}$.

Circle

The locus of points in a plane each of which is equidistant from a fixed point (the center); the constant distance is called the radius.

Tangent Line to a Circle

A line that intersects a circle at exactly one point (the point of tangency), perpendicular to the radius drawn to that point.

Parabola

The set of points in a plane that are equidistant from a fixed line called the directrix and a fixed point not on the line called the focus.

Latus Rectum of a Parabola

The chord passing through the focus perpendicular to the axis of symmetry of the parabola, having length equal to the focal width $4p$.

Ellipse

The locus of all points in a plane such that the sum of the distances from each point to two fixed points (foci) is constant ($2a$).

Eccentricity of an Ellipse

The ratio of the distance between the two foci to the length of the major axis, defined by $e = \frac{c}{a}$, where $0 < e < 1$.

Hyperbola

The set of all points in a plane such that the absolute difference of the distances from each point to two fixed points (foci) is constant ($2a$).

Eccentricity of a Hyperbola

The ratio of the distance between the foci to the length of the transverse axis, defined by $e = \frac{c}{a}$, where $e > 1$.

Conic Sections

Curves generated when a plane intersects a right circular double cone, forming a circle, ellipse, parabola, or hyperbola.