A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.
Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:
By a "set" we mean any collection M into a whole of definite, distinct objects m (which are called the "elements" of M) of our perception [Anschauung] or of our thought.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if they have the same elements.
Describing Sets
There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:
A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.
The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in brackets:
C = {4, 2, 1, 3}
D = {blue, white, red}
Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple. For example,
{6, 11} = {11, 6} = {11, 11, 6, 11},
because the extensional specification means merely that each of the elements listed is a member of the set.
Subsets
If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.
A is a subset of B
If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is proper superset of A).
Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).
Example:
1. The set of all men is a proper subset of the set of all people.
2. {1, 3} ⊊ {1, 2, 3, 4}.
3. {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
The empty set is a subset of every set and every set is a subset of itself:
1. ∅ ⊆ A.
2. A ⊆ A.
An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:
1. A = B if and only if A ⊆ B and B ⊆ A.
Power sets
The power set of a set S is the set of all subsets of S. This includes the subsets formed from all the members of S and the empty set. If a finite set S has cardinality n then the power set of S has cardinality 2n. The power set can be written as P(S).
If S is an infinite (either countable or uncountable) set then the power set of S is always uncountable. Moreover, if S is a set, then there is never a bijection from S onto P(S). In other words, the power set of S is always strictly "bigger" than S.
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 23 = 8. This relationship is one of the reasons for the terminology power set.
Cardinality
The cardinality | S | of a set S is "the number of members of S." For example, since the French flag has three colors, | B | = 3.
There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and is denoted by the symbol ∅ (other notations are used; see empty set). For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is important in mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.
Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
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. One of these is the empty set. Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include:
1. P, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
2. N, denoting the set of all natural numbers: N = {1, 2, 3, . . .}.
3. Z, denoting the set of all integers (whether positive, negative or zero): Z = {... , −2, −1, 0, 1, 2, ...}.
4. Q, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b : a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1.
4. R, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as π, e, and √2, as well as numbers that cannot be defined).
5. C, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.
6. H, denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.
Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related field.
Basic operations
There are several fundamental operations for constructing new sets from given sets.
1. Unions
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B.
The union of A and B, denoted A ∪ B
Examples:
1. {1, 2} ∪ {red, white} ={1, 2, red, white}.
2. {1, 2, green} ∪ {red, white, green} ={1, 2, red, white, green}.
3. {1, 2} ∪ {1, 2} = {1, 2}.
Some basic properties of unions:
1. A ∪ B = B ∪ A.
2. A ∪ (B ∪ C) = (A ∪ B) ∪ C.
3. A ⊆ (A ∪ B).
4. A ∪ A = A.
5. A ∪ ∅ = A.
6. A ⊆ B if and only if A ∪ B = B
2. Intersections
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. If A ∩ B = ∅, then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:
1. {1, 2} ∩ {red, white} = ∅.
2. {1, 2, green} ∩ {red, white, green} = {green}.
3. {1, 2} ∩ {1, 2} = {1, 2}.
Some basic properties of intersections:
1. A ∩ B = B ∩ A.
2. A ∩ (B ∩ C) = (A ∩ B) ∩ C.
3. A ∩ B ⊆ A.
4. A ∩ A = A.
5. A ∩ ∅ = ∅.
6. A ⊆ B if and only if A ∩ B = A.
3. Complements
Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B, (or A − B) is the set of all elements which are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.
In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.
The relative complement of B in A
The complement of A in U
The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.
Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:
By a "set" we mean any collection M into a whole of definite, distinct objects m (which are called the "elements" of M) of our perception [Anschauung] or of our thought.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if they have the same elements.
Describing Sets
There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:
A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.
The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in brackets:
C = {4, 2, 1, 3}
D = {blue, white, red}
Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple. For example,
{6, 11} = {11, 6} = {11, 11, 6, 11},
because the extensional specification means merely that each of the elements listed is a member of the set.
Subsets
If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.
A is a subset of B
If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is proper superset of A).
Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).
Example:
1. The set of all men is a proper subset of the set of all people.
2. {1, 3} ⊊ {1, 2, 3, 4}.
3. {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
The empty set is a subset of every set and every set is a subset of itself:
1. ∅ ⊆ A.
2. A ⊆ A.
An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:
1. A = B if and only if A ⊆ B and B ⊆ A.
Power sets
The power set of a set S is the set of all subsets of S. This includes the subsets formed from all the members of S and the empty set. If a finite set S has cardinality n then the power set of S has cardinality 2n. The power set can be written as P(S).
If S is an infinite (either countable or uncountable) set then the power set of S is always uncountable. Moreover, if S is a set, then there is never a bijection from S onto P(S). In other words, the power set of S is always strictly "bigger" than S.
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 23 = 8. This relationship is one of the reasons for the terminology power set.
Cardinality
The cardinality | S | of a set S is "the number of members of S." For example, since the French flag has three colors, | B | = 3.
There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and is denoted by the symbol ∅ (other notations are used; see empty set). For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is important in mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.
Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
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. One of these is the empty set. Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include:
1. P, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
2. N, denoting the set of all natural numbers: N = {1, 2, 3, . . .}.
3. Z, denoting the set of all integers (whether positive, negative or zero): Z = {... , −2, −1, 0, 1, 2, ...}.
4. Q, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b : a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1.
4. R, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as π, e, and √2, as well as numbers that cannot be defined).
5. C, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.
6. H, denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.
Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related field.
Basic operations
There are several fundamental operations for constructing new sets from given sets.
1. Unions
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B.
The union of A and B, denoted A ∪ B
Examples:
1. {1, 2} ∪ {red, white} ={1, 2, red, white}.
2. {1, 2, green} ∪ {red, white, green} ={1, 2, red, white, green}.
3. {1, 2} ∪ {1, 2} = {1, 2}.
Some basic properties of unions:
1. A ∪ B = B ∪ A.
2. A ∪ (B ∪ C) = (A ∪ B) ∪ C.
3. A ⊆ (A ∪ B).
4. A ∪ A = A.
5. A ∪ ∅ = A.
6. A ⊆ B if and only if A ∪ B = B
2. Intersections
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. If A ∩ B = ∅, then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:
1. {1, 2} ∩ {red, white} = ∅.
2. {1, 2, green} ∩ {red, white, green} = {green}.
3. {1, 2} ∩ {1, 2} = {1, 2}.
Some basic properties of intersections:
1. A ∩ B = B ∩ A.
2. A ∩ (B ∩ C) = (A ∩ B) ∩ C.
3. A ∩ B ⊆ A.
4. A ∩ A = A.
5. A ∩ ∅ = ∅.
6. A ⊆ B if and only if A ∩ B = A.
3. Complements
Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B, (or A − B) is the set of all elements which are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.
In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.
The relative complement of B in A
The complement of A in U
The symmetric difference of A and B
Examples:
1. {1, 2} \ {red, white} = {1, 2}.
2. {1, 2, green} \ {red, white, green} = {1, 2}.
3. {1, 2} \ {1, 2} = ∅.
4. {1, 2, 3, 4} \ {1, 3} = {2, 4}.
5. If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then E′ = O.
Some basic properties of complements:
1. A \ B ≠ B \ A.
2. A ∪ A′ = U.
3. A ∩ A′ = ∅.
4. (A′)′ = A.
5. A \ A = ∅.
6. U′ = ∅ and ∅′ = U.
7. \ B = A ∩ B′.
An extension of the complement is the symmetric difference, defined for sets A, B as
Examples:
1. {1, 2} \ {red, white} = {1, 2}.
2. {1, 2, green} \ {red, white, green} = {1, 2}.
3. {1, 2} \ {1, 2} = ∅.
4. {1, 2, 3, 4} \ {1, 3} = {2, 4}.
5. If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then E′ = O.
Some basic properties of complements:
1. A \ B ≠ B \ A.
2. A ∪ A′ = U.
3. A ∩ A′ = ∅.
4. (A′)′ = A.
5. A \ A = ∅.
6. U′ = ∅ and ∅′ = U.
7. \ B = A ∩ B′.
An extension of the complement is the symmetric difference, defined for sets A, B as
For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}.
Reference:
http://en.wikipedia.org/wiki/Set_(mathematics)
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