Mathematics Lecture: Numbers, Sets, Functions, and Geometry

Practice flashcards covering basic number systems, set theory, functions, coordinate geometry, and quadratic equations based on the lecture notes.

Natural Numbers

The set of numbers used to keep a count of objects, denoted as $\mathbb{N} = \{0, 1, 2, 3, 4, \dots\}$, where $0$ represents no objects at all.

Integers

The set of numbers including positive whole numbers, zero, and negative numbers, represented as $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Prime Number

A number $p$ that has exactly two factors: $\{1, p\}$. Note that $1$ is not a prime number because it only has one factor.

Prime Factorisation

The process of decomposing a number into a unique product of prime numbers, such as $12 = 2 \times 2 \times 3 = 2^2 \times 3$.

Rational Numbers

Numbers that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and the denominator $q \neq 0$, denoted by $\mathbb{Q}$.

Density

A property of rational and real numbers signifying that between any two numbers, another one can always be found (such as their average).

Real Numbers

The set of all rational and irrational numbers, denoted by $\mathbb{R}$, which correspond to all points on the number line.

Complex Numbers

A number system that extends real numbers to include values like $\sqrt{-1}$, denoted as $i$.

Cardinality

The number of items or elements contained within a set.

Russell's Paradox

A principle in set theory stating that the collection of all sets is not itself a set.

Power Set

The set containing all subsets of a given set $X$. A set with $n$ elements has a power set with $2^n$ subsets.

Set Comprehension

Also known as set builder form, it is a formal notation to define a set by applying a condition to each element of an existing set, such as $\{x \mid x \in \mathbb{Z}, x \pmod{2} = 0\}$.

Union

A set operation that combines all elements from two sets $X$ and $Y$, denoted as $X \cup Y$.

Intersection

A set operation that identifies only the elements common to both set $X$ and set $Y$, denoted as $X \cap Y$.

Cartesian Product

The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$, denoted as $A \times B$.

Equivalence Relation

A binary relation that is reflexive, symmetric, and transitive, used to partition a set into equivalence classes.

Domain

The set of all possible input values for a function.

Codomain

The set of all possible values that the output of a function can take.

Injective

A property of a function where different inputs produce different outputs, also known as a one-one mapping.

Surjective

A property of a function where the range is equal to the codomain, also known as onto.

Bijective

A function that is both injective and surjective, establishing a $1\text{-}1$ correspondence between the domain and codomain.

Slope

Also called the gradient, it is the rate of change defined as $m = \frac{y_1 - y_2}{x_1 - x_2} = \tan(\theta)$.

Section Formula

A formula used to find the coordinates of a point $P$ that divides a line segment $AB$ in the ratio $m:n$, given by $x = \frac{mx_2 + nx_1}{m+n}$ and $y = \frac{my_2 + ny_1}{m+n}$.

Parabola

The name of the graph resulting from any quadratic function of the form $f(x) = ax^2 + bx + c$.

Vertex

The point at which the axis of symmetry intersects a parabola, representing the minimum or maximum value of the quadratic function.

FOIL Method

A mnemonic for finding the product of two binomials by summing the products of the First, Outer, Inner, and Last terms.

Discriminant

The value $D = b^2 - 4ac$ used to determine the type and number of roots for a quadratic equation.

Polynomial

A mathematical expression that is a sum of several terms (monomials) where exponents of variables are natural numbers.