In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, especially in set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, a set 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. Although it was invented 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 A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B.
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Definitions
If A and B are sets and every element In mathematics, an element or member of a set is any one of the distinct objects that make up that set of A is also an element of B, then:
-
- A is a subset of (or is included in) B, denoted by ,
- or equivalently
- B is a superset of (or includes) A, denoted by
If A is a subset of B, but A is not equal Equality is the paradigmatic example of the more general concept of equivalence relations on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation on S that is to B (i.e. there exists at least one element of B not contained in A), then
-
- A is also a proper (or strict) subset of B; this is written as
- or equivalently
- B is a proper superset of A; this is written as
For any set S, the inclusion relation ⊆ is a partial order In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are on the set of all subsets of S (the power set In mathematics, given a set S, the power set of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set of S).
The symbols ⊂ and ⊃
Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively, instead of the symbols ⊆ and ⊇, but with the same meaning. So for example, for these authors, it is true of every set A that A ⊂ A.
Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, in place of and This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may be equal to y, or maybe not, but if x < y, then x definitely does not equal y, but is strictly less than y. Similarly, using the "⊂ means proper subset" convention, if A ⊆ B, then A may or may not be equal to B, but if A ⊂ B, then A is definitely not equal to B.
Examples
- The set {1, 2} is a proper subset of {1, 2, 3}.
- Any set is a subset of itself, but not a proper subset.
- The empty set In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set, denoted by ∅, is also a subset of any given set X. (This statement is vacuously true A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is false. For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room.) The empty set is always a proper subset, except of itself.
- The set {x: x is a prime number In mathematics, a prime number is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) are: greater than 2000} is a proper subset of {x: x is an odd number greater than 1000}
- The set of natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century is a proper subset of the set of rational numbers In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q , which stands for quotient and the set of points in a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge if they are adjacent vertices, or is a proper subset of the set of points in a line In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some. These are counter-intuitive examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole (see Cardinality of infinite sets In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers).
Other properties of inclusion
Inclusion is the canonical partial order In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. These relations are in the sense that every partially ordered set (X, ) is isomorphic In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f to some collection of sets ordered by inclusion. The ordinal numbers In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated are a simple example—if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].
For the power set In mathematics, given a set S, the power set of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set of a set S, the inclusion partial order is (up to an order isomorphism In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets . Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the) the Cartesian product In mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept of k = |S| (the cardinality In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating S = {s1, s2, …, sk} and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from {0,1}k of which the ith coordinate is 1 if and only if si is a member of T.
See also
References
- Jech, Thomas Thomas J. Jech is a set theorist who was at Penn State for more than 25 years. He was educated at Charles University (his adviser was Petr Vopěnka) and is now at the Institute of Mathematics of the Academy of Sciences of the Czech Republic (2002). Set Theory. Springer-Verlag. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 3-540-44085-2.
Categories: Basic concepts in set theory
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