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In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is ...
Set-builder notation. The set of all even integers, expressed in set-builder notation. In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
In mathematics, a transformation or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [2] [3] [4] Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine ...
Class (set theory) In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially ...
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the ...
Example: The blue circle represents the set of points (x, y) satisfying x 2 + y 2 = r 2. The red disk represents the set of points (x, y) satisfying x 2 + y 2 < r 2. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set. In mathematics, an open set is a generalization of an open ...
In mathematics, for a function , the image of an input value is the single output value produced by when passed . The preimage of an output value is the set of input values that produce . More generally, evaluating at each element of a given subset of its domain produces a set, called the " image of under (or through) ".
