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 intentional 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.
For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as:
{1, 2, 3, ..., 1000},
where the ellipsis ("...") indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.
The notation with braces may also be used in an intentional specification of a set. In this usage, the braces have the meaning "the set of all ...". So, E = {playing card suits} is the set whose four members are ♠, ♦, ♥, and ♣. A more general form of this is set-builder notation , through which, for instance, the set F of the twenty smallest integers that are four less than perfect squares can be denoted:
F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19}
In this notation, the colon (":") means "such that", and the description can be interpreted as "F is the set of all numbers of the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the vertical bar ("|") is used instead of the colon.
One often has the choice of specifying a set intentionally or extensionally. In the examples above, for instance, A = C and B = D.
Membership
Main article: Element (mathematics)
The key relation between sets is membership – when one set is an element of another. If a is a member of B, this is denoted a ∈ B, while if c is not a member of B then c ∉ B. For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,
4 ∈ A and 285 ∈ F; but
9 ∉ F and green ∉ B.
Subsets
Main article: Subset
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.
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:
• The set of all men is a proper subset of the set of all people.
• {1, 3} ⊊ {1, 2, 3, 4}.
• {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:
• ∅ ⊆ A.
• A ⊆ A.
An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:
• A = B if and only if A ⊆ B and B ⊆ A.
Power sets
Main article: Power set
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
Main article: 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:
• P, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
• N, denoting the set of all natural numbers: N = {1, 2, 3, . . .}.
• Z, denoting the set of all integers (whether positive, negative or zero): Z = {... , −2, −1, 0, 1, 2, ...}.
• 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.
• 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).
• C, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.
• 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 fields.
الأربعاء 22 ديسمبر 2010 - 18:08
