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Sets, Relations and Functions
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
As mentioned before, the sets in the same equivalence class are said to have the same cardinality or the cardinal number. Intuitively it must be clear that equipotent sets have the same “number” of elements. The cardinal number of any finite set is a positive integer, while the cardinal numbers of infinite sets are denoted by certain symbols. The cardinal number of the infinite set N $ \mathbf N $ (the set of positive integers) is denoted by ℵ0 $ {\aleph }_0 $ (aleph not). ℵ $ \aleph $ is the first character of the Hebrew language.
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The natural idea of counting that lets us compare finite sets (two sets are “equivalent” if they have the same number of elements) may be generalized to the case of infinite sets. Every set may be assigned a symbol, called its “cardinal number,” which describes its “number of elements” in the sense that, indeed, in the case of a finite set, its cardinal number is equal to its number of elements.
Control charts for monitoring relative risk rate in the presence of Weibull competing risks with censored and masked data
Published in Quality Technology & Quantitative Management, 2023
Adel Ahmadi Nadi, Robab Afshari, Bahram Sadeghpour Gildeh
where and are the p.d.f. (s.f.) of and , respectively. These functions can be obtained by substituting the corresponding parameters in (1) and (2), respectively. In addition, represents the set of observations with observed or imputed causes that the item fails due to the primary cause, is the set of observations with observed or imputed causes where the failure is owned by the competing cause, and denotes the set of censored observations. It is concluded that and hence and where shows the cardinal number of set . Substituting , and into (13), the likelihood function becomes
Accuracy of the equilibrium structure of sulphur dioxide
Published in Molecular Physics, 2022
First, the CCSD(T) structure was estimated at infinite basis size by extrapolating the aug-cc-pwCVQZ and aug-cc-pwCV5Z results with the following empirical equation [44] where n indicates the cardinal number of the larger basis set. This extrapolation formula is known to have an accuracy of about 0.001 Å when only first-row atoms are involved, at least for covalent bonds. In the case of SO2, the accuracy of the extrapolation could be worse. To confirm the correct order of magnitude of this correction, the structure was also optimised at the CCSD(T) level of theory with the aug-cc-pV5Z and aug-cc-pV6Z basis sets. The correction (aug-cc-pV5Z → aug-cc-pV6Z) is Δr = −0.00171 Å and Δθ = +0.073°. These corrections confirm the validity of the extrapolation.
Subsets of fields whose nth-root functions are rational functions
Published in International Journal of Mathematical Education in Science and Technology, 2018
The following observation is in the spirit of the ninth sentence of the Introduction. Let F be an ordered field with its set of positive elements. Suppose that n is even. Let Λn denote the set of nth-roots of 1 in F. As 1, −1 ∈ Λn, we have 2 ≤ |Λn| ≤ n. Then, all possible nth-root functions ψ on Fn with values in F can be obtained as follows. For each u ∈ Fn, choose wu ∈ Λn at random and (using Proposition 2.1(b)) let vu denote the unique element of such that u = vnu. Then, ψ can be given by the assignment u ↦ wuvu for one of the random choices of the elements wu. Thus, the cardinal number of the set of nth-root functions on Fn with values in F is . This is an infinite cardinal number, since |Λn| ≥ 2 and Fn is infinite. Hence, by Proposition 2.1(c), the cardinal number of the set of nth-root functions on Fn with values in F is |Λn||F| (>|F|).By Remark 2.1(a), the hypothesis that n ≥ 2 in Proposition 2.1(d) cannot be deleted. Observe also that the above proof of Proposition 2.1(d) would not apply if ψ is the identity function on F, for then n = 1 and m = 1, so that the polynomial mnXn − X would be the zero polynomial, not a polynomial of degree n.