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Projection (mathematics) – Mapping equal to its square under mapping composition; Projection (measure theory) Projection (linear algebra) – Idempotent linear transformation from a vector space to itself; Projection (relational algebra) – Operation that restricts a relation to a specified set of attributes
A projection may also refer to a mapping which has a right inverse. Both notions are strongly related, as follows. Let p be an idempotent mapping from a set A into itself (thus p ∘ p = p) and B = p(A) be the image of p. If we denote by π the map p viewed as a map from A onto B and by i the injection of B into A (so that p = i ∘ π), then ...
Every set is a projective object in Set (assuming the axiom of choice). The finitely presentable objects in Set are the finite sets. Since every set is a direct limit of its finite subsets, the category Set is a locally finitely presentable category. If C is an arbitrary category, the contravariant functors from C to Set are often an important ...
If S is indexed by a set I consisting of all the natural numbers N or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) f : F S → N from the free group on S to the natural numbers, such that we can find algorithms that, given f(w), calculate w, and vice versa.
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.
Cardinal functions are widely used in topology as a tool for describing various topological properties. [4] [5] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions ...
And we can map the powerset of into the Cantor set, a subset of the real numbers. So statements about H ℵ 1 {\displaystyle H_{\aleph _{1}}} can be converted into statements about the reals. Therefore, H ℵ 1 ⊂ L ( R ) {\displaystyle H_{\aleph _{1}}\subset L(R)} , where L ( R ) is the smallest transitive inner model of ZF containing all the ...
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