In mathematics, a well-defined class of elements, ie a class where it is possible to tell exactly whether any one element does or does not belong to it. We can have the set of all even numbers, as every number is either even or not even, but we cannot have the set of all large numbers, as we do not know what is meant by large. The empty set ? is the set with no elements. The universal set ? or ? is the set of all elements, and the complement A? of a set A is the set of all elements in ? which are not in A. However, universal sets must also be defined carefully, else paradoxes result; one cannot speak of the set of all sets, for example. To do so would admit Russell's paradox (based on the set of all sets that are not members of themselves) which destroyed Frege's attempt to base all mathematics on logic. The intersection of two sets A and B (written A?B) is the set of all elements in both A and B. The union of two sets A and B (written A?B) is the set of all elements in either A or B or both.
This article is about sets in mathematics. Though a simple idea, it is nevertheless one of the most important and fundamental concepts in modern mathematics, and the study of the structure of possible sets, set theory, is quite rich.This article gives a brief and basic introduction to what mathematicians call "intuitive" or "naive" set theory; For a rigorous modern axiomatic treatment of sets see axiomatic set theory. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Two sets A and B are said to be equal, written A = B, if they have the same members. All set operations preserve the property that each element in the set is unique.
Describing sets
Not all sets have precise descriptions of any sort;
Some sets may be described in words, for example:
A is the set whose members are the first four positive whole numbers.By convention, a set can also be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces), for example:
C = {4, 2, 1, 3} D = {red, white, blue}Two different descriptions may define the same set.
Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list.
Similarly the set of even numbers can be described by the notation:
{2, 4, 6, 8, ... For example the set F, whose members are the first twenty numbers which are four less than a square integer, can be described using the following:F = {n2 – 4 : n is an integer; and 0 ≤ n ≤ 19}
In this description, the colon (:) means "such that", and the mathematician interprets this description as "F is the set of numbers of the form n2 – 4, such that n is a whole number in the range from 0 to 19 inclusive."
Set membership
If something is or is not an element of a particular set then this is symbolised by and respectively.
Cardinality of a set
Each of the sets described above has a definite number of members; for example, the set A has four members, while the set B has three members. For example, the set A of all three-sided squares has zero members, and thus A = ø.
Subsets
If every member of the set A is also a member of the set B, then A is said to be a subset of B, written , also pronounced A is contained in B.
A is a subset of BExamples:
The set of all men is a proper subset of the set of all people.The empty set is a subset of every set and every set is a subset of itself:
Special sets
There are some sets which hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. Some special sets of numbers include:
denotes the set of all primes. denotes the set of all rational numbers (that is, the set of all proper and improper fractions).Each of these sets of numbers has infinite size, though , although the primes are generally used less than the others outside of number theory and related fields.
Unions
There are several ways to construct new sets from existing ones. The union of A and B, denoted by A U B, is the set of all things which are members of either A or B.
The union of A and BExamples:
{1, 2} U {red, white} = {1, 2, red, white} {1, 2, green} U {red, white, green} = {1, 2, red, white, green} {1, 2} U {1, 2} = {1, 2}Some basic properties of unions:
A U B = B U A A is a subset of A U B A U A = A A U ø = AIntersections
A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B.
The intersection of A and BExamples:
{1, 2} ∩ {red, white} = ø {1, 2, green} ∩ {red, white, green} = {green} {1, 2} ∩ {1, 2} = {1, 2}Some basic properties of intersections:
A ∩ B = B ∩ A A ∩ B is a subset of A A ∩ A = A A ∩ ø = øComplements
Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B − A, (or B \ A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing green from {1,2,3};
In certain settings all sets under discussion are considered to be subsets of a given universal set U.
The relative complementof A in B The complement of A in U
Examples:
{1, 2} − {red, white} = {1, 2} {1, 2, green} − {red, white, green} = {1, 2} {1, 2} − {1, 2} = ø If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E′ = O.Some basic properties of complements:
A U A′ = U A ∩ A′ = ø (A′ )′ = A A − A = ø A − B = A ∩ B′Further reading
For more information on the basic properties of sets, subsets, intersections, unions and complements, see algebra of sets. For a more general development of these ideas and others in set theory, see naive set theory.
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