Search results
Results from the WOW.Com Content Network
1. Naive set theory can mean set theory developed non-rigorously without axioms 2. Naive set theory can mean the inconsistent theory with the axioms of extensionality and comprehension 3. Naive set theory is an introductory book on set theory by Halmos natural The natural sum and natural product of ordinals are the Hessenberg sum and product NCF
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 ...
Axiom schema of replacement: the image [] of the domain set under the definable class function is itself a set, . Suppose P {\displaystyle P} is a definable binary relation (which may be a proper class ) such that for every set x {\displaystyle x} there is a unique set y {\displaystyle y} such that P ( x , y ) {\displaystyle P(x,y)} holds.
More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I {\displaystyle I} , known as the index set, to F {\displaystyle F} , in which case the sets of the family are indexed by members of I {\displaystyle I} . [ 1 ]
The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set. Formally, let φ {\displaystyle \varphi } be any formula in the language of ZFC whose free variables are among x , y , A , w 1 , … , w n , {\displaystyle x,y,A,w_{1},\dotsc ,w_{n},} so that in particular B {\displaystyle B ...
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.