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That is, the power set ℘ of a finite set S is finite, with cardinality | |. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do ...
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more refined notions. A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y.
The finite intersection property, abbreviated FIP, says that the intersection of any finite number of elements of a set is non-empty first 1. A set of first category is the same as a meager set: one that is the union of a countable number of nowhere-dense sets. 2. An ordinal of the first class is a finite ordinal 3.
In mathematics education, Finite Mathematics is a syllabus in college and university mathematics that is independent of calculus. A course in precalculus may be a prerequisite for Finite Mathematics.
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line {}, which consists of the real numbers and .
A set of real numbers (hollow and filled circles), a subset of (filled circles), and the infimum of . Note that for totally ordered finite sets, the infimum and the minimum are equal. A set A {\displaystyle A} of real numbers (blue circles), a set of upper bounds of A {\displaystyle A} (red diamond and circles), and the smallest such upper ...