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In mathematics, a set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. A finite set is a collection of a finite number of objects; the alternative is an infinite set. For a discussion of the properties and axioms concerning the construction of sets, see naive set theory and axiomatic set theory. Here we give only a brief overview of the concept.
Sets are one of the basic concepts of mathematics. A set is, more or less, just a collection of objects, called its elements. Standard notation uses braces around the list of elements, as in:
All three lines above denote the same set. As you see, it is possible to describe one and the same set in different ways: either by listing all its elements (best for small finite sets) or by giving a defining property of all its elements; and it does not matter in what order, or how many times, the elements are listed, if a list is given.
If <math>A</math> and <math>B</math> are sets and every <math>x</math> in <math>A</math> is also contained in <math>B</math>, then <math>A</math> is said to be a subset of <math>B</math>, denoted <math>A \subseteq B</math>. If atleast one element in <math>B</math> is not also in <math>A</math>, <math>A</math> is called a proper subset of <math>B</math>, denoted <math>A \subset B</math>. Every set has as subsets itself, called the improper subset, and the empty set {} or <math>\emptyset</math>. The fact that an element <math>x</math> belongs to the set <math>A</math> is denoted <math>x \in A</math>.
The union of a collection of sets <math>S = {S_1, S_2, S_3, \cdots}</math> is the set of all elements contained in at least one of the sets <math>S_1, S_2, S_3, \cdots</math>
The intersection of a collection of sets <math>T = {T_1, T_2, T_3, \cdots}</math> is the set of all elements contained in all of the sets.
These unions and intersections are denoted
and
respectively.
The "number of elements" in a certain set is called the cardinal number of the set and denoted <math>A</math> for a set <math>A</math> (for a finite set this is an ordinary number, for an infinite set it differentiates between different "degrees of infiniteness", named <math>\aleph_0</math> (aleph zero), <math>\aleph_1, \aleph_2 ...</math>).
The set of all subsets of <math>X</math> is called its power set and is denoted <math>2^X</math> or <math>P(X)</math>. This power set is a Boolean algebra under the operations of union and intersection.
The set of functions from a set A to a set B is sometimes denoted by B^{A}. It is a generalisation of the power set in which 2 could be regarded as the set {0,1} (see natural number).
The cartesian product of two sets A and B is the set
The sum of two sets A and B is the set
Special Remarks About Terminology
Care must be taken with verbal descriptions of sets. One can describe in words a set whose existence is paradoxical. If one assumes such a set exists, an apparent paradox or antinomy may occur. Axiomatic set theory was created to avoid these problems.
For example, suppose we call a set "wellbehaved" if it doesn't contain itself as an element. Now consider the set S of all wellbehaved sets. Is S itself wellbehaved? There is no consistent answer; this is Russell's paradox. In axiomatic set theory, the set S is not allowed, and we have no paradox.
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