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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 ...
In the examples below, the Euler diagram depicts that the sets Animal and Mineral are disjoint since the corresponding curves are disjoint, and also that the set Four Legs is a subset of the set of Animals. The Venn diagram, which uses the same categories of Animal, Mineral, and Four Legs, does not encapsulate
A Venn diagram is a representation of mathematical sets: a mathematical diagram representing sets as circles, with their relationships to each other expressed through their overlapping positions, so that all possible relationships between the sets are shown.
Euler diagram of number sets: Image title: Euler diagram of selected number sets (assuming transcendental numbers are real) by CMG Lee. Bold numbers denote only numbers in their regions. Italics denote examples in each region. In the SVG file, hover over a set to highlight it. Width: 100%: Height: 100%
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
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".
The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra.
Functional and not injective. For example, the red relation in the diagram is many-to-one, but the green, blue and black ones are not. Many-to-many [d] Not injective nor functional. For example, the black relation in the diagram is many-to-many, but the red, green and blue ones are not. Uniqueness and totality properties: A function [d]