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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, ...
In logic, a set of symbols is commonly used to express logical representation. ... logical (inclusive) disjunction: or propositional logic, Boolean algebra:
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 logic, disjunction, also known as logical disjunction or logical or or logical addition or inclusive disjunction, is a logical connective typically notated as and read aloud as "or".
In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
Inclusion logic and exclusion logic are obtained by adding only inclusion atoms or exclusion atoms to first-order logic, respectively. Inclusion logic sentences correspond in expressive power to greatest fixed-point logic sentences; hence inclusion logic captures (least) fixed-point logic on finite models, and PTIME over finite ordered models. [29]
In set theory, each line represents a set instead of a logical statement; A replaces p and B replaces q. When used for sets, a dot above the line represents inclusion, where a dot below represents exclusion. As in logic, basic set operations can be represented visually using R-diagrams:
Inclusion maps are seen in algebraic topology where if is a strong deformation retract of , the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence). Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds.