The language of set

Set

A collection of well-defined objects, where the objects are called elements or members of the set.

Elements (Members)

The individual objects that belong to a set.

Element Symbol (∈)

Denotes that an object is an element of a set.

Not an Element Symbol (∉)

Denotes that an object is not an element of a set.

Braces and Commas

Symbols used to enclose and separate the elements of a set.

Origin of Set Theory

Introduced as a formal mathematical term in 1879 by Georg Cantor.

Ellipsis Notation (…)

A variation of notation used to describe very large or infinite sets.

1. Roster Method / Listing Method / Tabular Method / Enumeration Method, 2. Set Builder Notation / Rule Method, 3. Interval Notation.

Three Ways of Describing a Set

Roster Method (Listing / Tabular / Enumeration)

A method where the elements of the set are enumerated, listed inside braces, and separated by commas.

Set Builder Notation (Rule Method)

A method that characterizes all elements in a set by stating the property or properties they must have, written as {x | P(x)}.

Interval Notation

A method used to describe the set of real numbers in an interval, consisting of 9 different notations.

Rational Numbers

The set of all terminating and repeating decimals.

Unit Set

A set that contains only one element.

Empty Set (Null Set)

A set containing no elements, denoted by { } or ∅.

Finite Set

A set whose elements can be counted and has a definite number of elements.

Infinite Set

A set whose elements cannot be counted because they go on indefinitely.

Cardinal Number

The number of distinct elements in a given set, denoted by n(A).

Equal Sets

Sets that contain exactly the same elements.

Equivalent Sets

Sets that have the same cardinal number or number of elements.

Universal Set

A set that contains all elements under consideration in a given context, denoted by U.

Joint Sets

Sets that have at least one common element.

Disjoint Sets

Sets that have no elements in common.

Subset (⊆)

Set A is a subset of B if and only if all elements of A are also elements of B.

Proper Subset

Set A is a proper subset of B if A is a subset of B and A is not equal to B.

Set Operations

Operations performed on sets, including Intersection, Union, Complement, Difference, and Product.

Intersection of Sets (∩)

The set containing all elements that are common to both set A and set B (A ∩ B).

Union of Sets (∪)

The set of elements that belong to either set A or set B or to both, written as A ∪ B = {x | x ∈ A or x ∈ B}.

Inclusive "Or" in Union

Means that x is an element of A, or B, or both; union is written without repeating elements.

Complement of a Set (A')

The set of all elements in the universal set U that are not elements of set A.

Difference of Two Sets (A - B)

The set of all elements of set A that are not elements of set B.

Cartesian Product (Cross Product, A × B)

The set of all possible ordered pairs (a, b) where a ∈ A and b ∈ B.

Venn Diagram

A pictorial representation of sets within an enclosing rectangle representing the universal set U, using geometric figures (circles, squares) to show subsets and overlapping regions for common areas.