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In mathematics, if is a subset of , then the inclusion map is the function that sends each element of to , treated as an element of ::, =. An inclusion map may also be referred to as an inclusion function , an insertion , [ 1 ] or a canonical injection .
In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within a second object. [ 1 ] For example, if m and n are two logical matrices , then
In a very abstract setting, the principle of inclusion–exclusion can be expressed as the calculation of the inverse of a certain matrix. [5] This inverse has a special structure, making the principle an extremely valuable technique in combinatorics and related areas of mathematics. As Gian-Carlo Rota put it: [6]
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).
The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }. Some useful properties of the inclusion relation are:
In mathematics, a submanifold of a manifold is a subset which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required.
Inclusion (set theory), or subset; Inclusion (Boolean algebra), the Boolean analogue to the subset relation; Inclusion map, or inclusion function, or canonical injection; Inclusion (logic), the concept that all the contents of one object are also contained within a second object
The notation [,) is used to indicate an interval from a to c that is inclusive of —but exclusive of . That is, [ 5 , 12 ) {\displaystyle [5,12)} would be the set of all real numbers between 5 and 12, including 5 but not 12.