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, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection In mathematics, a bijection, or a bijective function, is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y and no unmapped element exists in either X or Y f: X → Y such that both f and its inverse are monotone (order preserving). (In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse.)

For example, the set of integers and the set of even integers have the same order type, because the mapping preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) are not order isomorphic, because, even though the sets are the same size In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor. The elements of a countable set can be counted one at a time — although the counting may never finish, every element of the set will, there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positive integers (which has a least element), and that of negative integers (which has a greatest element). The open interval (0,1) of rationals is order isomorphic to the rationals ( provides a monotone bijection from the former to the latter); the half-closed intervals [0,1) and (0,1], and the closed interval [0,1], are three additional order type examples.

Since order-equivalence is an equivalence relation In mathematics, an equivalence relation is, loosely, a relation that specifies how to partition a set such that every element of the set is in exactly one of the partitions and the union of all the partitions equals the original set. Two elements of the set are considered equivalent if and only if they are elements of the same partition, it partitions the class of all ordered sets into equivalence classes.

Contents

Order type of well-orderings

Every well-ordered set is order-equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives Canonical is an adjective derived from canon. Canon comes from the Greek word kanon, "rule" , and is used in various meanings of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. For example, the order type of the natural numbers is ω.

The order type of a well-ordered set V is sometimes expressed as ord(V).[1]

For example, consider the set of even ordinals less than ω·2+7, which is:

V = {0, 2, 4, 6, ...; ω, ω+2, ω+4, ...; ω·2, ω·2+2, ω·2+4, ω·2+6}.

Its order type is:

ord(V) = ω·2+4 = {0, 1, 2, 3, ...; ω, ω+1, ω+2, ...; ω·2, ω·2+1, ω·2+2, ω·2+3}.

Because there are 2 separate lists of counting and 4 in sequence at the end.

Rational numbers

Any countable totally ordered set can be mapped injectively into the rational numbers in an order-preserving way.

Notation

The order type of the rationals 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 is usually denoted η. If a set S has order type σ, the order type of the dual In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop iff y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping of S (the reversed order) is denoted σ * .

See also

External links

References

  1. ^ Ordinal Numbers and Their Arithmetic

Categories: Ordinal numbers

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