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The following construction of the Vitali set shows one way that the axiom of choice can be used in a proof by transfinite induction: First, well-order the real numbers (this is where the axiom of choice enters via the well-ordering theorem ), giving a sequence r α ∣ α < β {\displaystyle \langle r_{\alpha }\mid \alpha <\beta \rangle ...
1. The domain is the real line .The set-family contains all the half-lines (rays) from a given number to positive infinity, i.e., all sets of the form {>} for some .For any set of real numbers, the intersection contains + sets: the empty set, the set containing the largest element of , the set containing the two largest elements of , and so on.
The cumulative hierarchy is a collection of sets V α indexed by the class of ordinal numbers; in particular, V α is the set of all sets having ranks less than α. Thus there is one set V α for each ordinal number α. V α may be defined by transfinite recursion as follows: Let V 0 be the empty set::=.
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 ]
Monotone class theorem for functions — Let be a π-system that contains and let be a collection of functions from to with the following properties: If A ∈ A {\displaystyle A\in {\mathcal {A}}} then 1 A ∈ H {\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}} where 1 A {\displaystyle \mathbf {1} _{A}} denotes the indicator function of A ...
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers.
In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
There are several methods of finding an optimal design, given an a priori restriction on the number of experimental runs or replications. Some of these methods are discussed by Atkinson, Donev and Tobias and in the paper by Hardin and Sloane. Of course, fixing the number of experimental runs a priori would be impractical. Prudent statisticians ...