Search results
Results From The WOW.Com Content Network
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.
For example, "1 < 3", "1 is less than 3", and "(1,3) ∈ R less" mean all the same; some authors also write "(1,3) ∈ (<)". Various properties of relations are investigated. A relation R is reflexive if xRx holds for all x, and irreflexive if xRx holds for no x. It is symmetric if xRy always implies yRx, and asymmetric if xRy implies that yRx ...
The combined region of the two sets is called their union, denoted by A ∪ B, where A is the orange circle and B the blue. The union in this case contains all living creatures that either are two-legged or can fly (or both). The region included in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B.
The open sets in the product topology are unions (finite or infinite) of sets of the form , where each U i is open in X i and U i ≠ X i only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the X i gives a basis for the product ∏ i ∈ I X i ...
If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. [11] In symbols: x ∈ ⋃ M ∃ A ∈ M , x ∈ A . {\displaystyle x\in \bigcup \mathbf {M} \iff \exists A\in \mathbf {M} ,\ x\in A.}
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. [4]
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals" Euler diagram showing the relationships between different Solar System objects
The image of the function is the set of all output values it may produce, that is, the image of . The preimage of f {\displaystyle f} , that is, the preimage of Y {\displaystyle Y} under f {\displaystyle f} , always equals X {\displaystyle X} (the domain of f {\displaystyle f} ); therefore, the former notion is rarely used.