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Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite. If an infinite set is a well-ordered ...
As an example of an application of AC ω, here is a proof (from ZF + AC ω) that every infinite set is Dedekind-infinite: [2] Let X {\displaystyle X} be infinite. For each natural number n {\displaystyle n} , let A n {\displaystyle A_{n}} be the set of all n {\displaystyle n} -tuples of distinct elements of X {\displaystyle X} .
The element of value is compatible with the term luminosity, and can be "measured in various units designating electromagnetic radiation". [6] The difference in values is often called contrast , and references the lightest (white) and darkest (black) tones of a work of art, with an infinite number of grey variants in between. [ 6 ]
In the formal language of the Zermelo–Fraenkel axioms, the axiom is expressed as follows: [2] ( ( ()) ( ( (( =))))). In technical language, this formal expression is interpreted as "there exists a set 𝐼 (the set that is postulated to be infinite) such that the empty set is an element of it and, for every element of 𝐼, there exists an element of 𝐼 consisting of just the elements of ...
An infinite set is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
Cantor's diagonal argument (among various similar names [note 1]) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.