Introduction to Sets and the Real Number System

Comprehensive vocabulary flashcards covering the real number system, set definitions, representations, types, and operations based on the lecture notes.

Natural Numbers

The set of positive counting numbers starting from 1, represented as $1, 2, 3, 4, 5, 6, \dots$.

Whole Numbers

The set of numbers starting from 0, represented as $0, 1, 2, 3, 4, 5, 6, \dots$.

Integers

The set containing whole numbers and their negative counterparts, represented as $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$.

Rational Numbers

Numbers including recurring or terminating decimals and simple fractions, such as $-1.5, -1, -0.5, 0, 0.5, 1, 1.5, \dots$.

Irrational Numbers

Numbers that cannot be written as a simple fraction, including values like $\sqrt{2}, \sqrt{3}, 1, \sqrt{11}$.

Real Numbers

The set of all rational and irrational numbers combined, including examples like $-2, -1.816, 0, 1, \sqrt{2}, \dots$.

Set

A collection of well-defined, distinct objects or elements that does not change from person to person, usually denoted by a capital letter.

Statement Form

A method of representing a set by describing its elements in words, for example: "The set of even numbers less than 10."

Roster or Listing Form

A method of representing a set by listing all elements separated by commas within curly braces $\text{\{ \}}$, such as $B = \{2, 4, 6, 8\}$.

Set-Builder Form (Rule Form)

A method describing the properties that the elements must satisfy, denoted as $A = \{x : \text{property}\}$ or $B = \{x \mid x \text{ is an even number less than 10}\}$.

Cardinality

The number of elements contained inside a set, denoted as $n(A)$.

Empty Set (Void or Null Set)

A set that does not contain any element, denoted by $\text{\{ \}}$ or $\emptyset$.

Singleton Set

A set that contains exactly one single element, such as $A = \{3\}$.

Finite Set

A set that consists of a definite, countable number of elements, such as natural numbers up to 10.

Infinite Set

A set that is not finite and has endless elements, such as the set of all natural numbers $A = \{1, 2, 3, \dots\}$.

Equivalent Sets

Two different sets $A$ and $B$ that have the same number of elements ($n(A) = n(B)$), represented as $A \approx B$.

Equal Sets

Sets that contain exactly the same elements, regardless of the order, such that $A = B$.

Disjoint Sets

Two sets $A$ and $B$ are called disjoint if they do not contain any common elements.

Subset

A set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$, denoted as $A \subseteq B$. Note that every set is a subset of itself and the null set is a subset of every set.

Proper Subset

If $A \subseteq B$ and $A \neq B$, then $A$ is a proper subset of $B$, denoted as $A \subset B$.

Superset

A set $A$ is the superset of $B$ if all elements of set $B$ are also elements of set $A$, represented as $A \supset B$.

Universal Set

The set of all possible values relevant to a particular condition, denoted by $U$.

Union of Sets

The set denoted as $A \cup B$ that contains all elements from both set $A$ and set $B$.

Intersection of Sets

The set denoted as $A \cap B$ that contains only the elements common to both set $A$ and set $B$.

Difference of Sets

The set denoted as $A - B$ that contains elements in set $A$ that are not present in set $B$.

Complement of Sets

The set of all elements in the universal set $U$ that are not present in set $A$, denoted as $A'$ (A prime).

Cartesian Product

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

Law of Double Complement

The set property stating that the complement of a complement returns the original set: $(P')' = P$.

Laws of Empty and Universal Set Complements

The properties stating that $\emptyset' = U$ and $U' = \emptyset$.