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  2. Subset - Wikipedia

    en.wikipedia.org/wiki/Subset

    These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...

  3. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. [19]

  4. Positive real numbers - Wikipedia

    en.wikipedia.org/wiki/Positive_real_numbers

    In mathematics, the set of positive real numbers, ... is the subset of those real numbers that are greater than zero. The non-negative real numbers, ...

  5. Dense set - Wikipedia

    en.wikipedia.org/wiki/Dense_set

    In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine ...

  6. Interval (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Interval_(mathematics)

    An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers. [1] If the infimum does not exist, one says often that the corresponding endpoint is .

  7. Construction of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Construction_of_the_real...

    An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...

  8. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    As an example, let be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from together with the set of integers and the set of positive real numbers +, ordered by subset inclusion as above.

  9. Lebesgue measure - Wikipedia

    en.wikipedia.org/wiki/Lebesgue_measure

    The first part of the definition states that the subset of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals I {\displaystyle I} covers E {\displaystyle E} in a sense, since the union of these intervals contains E {\displaystyle E} .