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If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. [3] Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an
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
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers , and the set of all subsets of the set of natural numbers.
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
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:
For example, one infinity—the one most people are familiar with—is an infinite set of natural numbers: 1, 2, 3, and so on. However, there’s also an infinite set of real numbers, which ...
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 ...
For example, the sets = {,,} and = {,,} are the same size as they each contain 3 elements. Beginning in the late 19th century, this concept was generalized to infinite sets , which allows one to distinguish between different types of infinity, and to perform arithmetic on them.