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In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection of subsets of a given set is called a family of subsets of , or a family of sets over .
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 ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
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 .
In mathematics, the infimum (abbreviated inf; pl.: infima) of a subset of a partially ordered set is the greatest element in that is less than or equal to each element of , if such an element exists. [1]
Equivalently, a subset of R n is open if every point in is the center of an open ball contained in . An example of a subset of R that is not open is the closed interval [0,1], since neither 0 - ε nor 1 + ε belongs to [0,1] for any ε > 0, no matter how small.
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 ...