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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 ...
The following definition translates this to any category. A concrete category is a category that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that is, an object in Set), which will be the basis of the free object to be
The free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
Arbeit macht frei ([ˈaʁbaɪt ˈmaxt ˈfʁaɪ] ⓘ) is a German phrase translated as "Work makes one free" or more idiomatically "Work sets you free" or "work liberates". The phrase originates from the 1873 novel Die Wahrheit macht frei ("The truth sets free") by Lorenz Diefenbach , a pastor and philologist, itself being a reference to John 8: ...
Set (abstract data type), a data type in computer science that is a collection of unique values Set (C++), a set implementation in the C++ Standard Library Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems
In computer science, a set is an abstract data type that can store unique values, without any particular order. It is a computer implementation of the mathematical concept of a finite set. Unlike most other collection types, rather than retrieving a specific element from a set, one typically tests a value for membership in a set.
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 ...
In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters , which are elements of the domain that can be referenced in the formula defining the relation.