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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 ...
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself.
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.
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. [3] Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language.
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".
If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers.
In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of . It is usually denoted by S ′ . {\displaystyle S'.} The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line .