Definitions, Types, Operations, and Visualizations
A set is a well-defined collection of distinct objects, considered as an object in its own right.
The objects that make up a set are called its elements or members. These can be anything: numbers, letters, or even other sets.
A set must be 'well-defined,' meaning there's a clear criterion for determining whether an object belongs to the set.
Examples include: the set of vowels in the English alphabet, the set of even numbers less than 10, or the set of planets in our solar system.
Sets are fundamental in mathematics and computer science, providing a basis for logic, relations, functions, and more.
Sets can be described by listing their elements within curly braces {}. For example, {1, 2, 3, 4, 5}.
Set-builder notation defines a set by specifying a property that its elements must satisfy. For example, {x | x is an even number}.
Sets can also be described using words. For example, 'the set of all prime numbers less than 20'.
The method of describing a set depends on the context and the nature of the set's elements. For finite sets, the roster method is often used.
Roster: {2, 3, 5, 7, 11, 13, 17, 19}. Set-builder: {x | x is a prime number less than 20}.
A set that contains no elements, denoted by {} or ∅.
A set with a finite number of elements. The number of elements is a non-negative integer.
A set with an infinite number of elements. It is not possible to count the elements.
A set containing only one element. For example, {5}.
A set containing all possible elements under consideration in a given context. It is typically denoted by U.
Venn diagrams are graphical representations of sets, using circles or other shapes to depict sets and their relationships.
Each circle represents a set, and the overlapping areas represent the intersection of those sets.
The universal set is typically represented by a rectangle enclosing all the circles.
Venn diagrams are used to illustrate set operations like union, intersection, and complement.
Two overlapping circles can represent the intersection and union of two sets A and B.
Intervals are a way of representing a continuous set of real numbers within a specified range.
An open interval (a, b) includes all numbers between a and b, but not a and b themselves.
A closed interval [a, b] includes all numbers between a and b, as well as a and b themselves.
Half-open intervals include one endpoint but not the other, such as (a, b] or [a, b).
Intervals can extend to infinity, such as (a, ∞) or (-∞, b]. These intervals are always open at the infinity end.
The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in A, or in B, or in both.
A ∪ B = {x | x ∈ A or x ∈ B}
If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
The union is represented by shading the entire area covered by both circles (sets A and B).
Union is commutative (A ∪ B = B ∪ A) and associative ((A ∪ B) ∪ C = A ∪ (B ∪ C)).
The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are in both A and B.
A ∩ B = {x | x ∈ A and x ∈ B}
If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.
The intersection is represented by the overlapping area between the two circles (sets A and B).
Intersection is commutative (A ∩ B = B ∩ A) and associative ((A ∩ B) ∩ C = A ∩ (B ∩ C)).
The difference of two sets A and B, denoted by A - B (or A \ B), is the set of all elements that are in A but not in B.
A - B = {x | x ∈ A and x ∉ B}
If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}.
The complement of a set A (with respect to the universal set U), denoted by A', is the set of all elements in U that are not in A.
A' = {x | x ∈ U and x ∉ A}