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Algebraic Aspects
Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019
Marlos A. G. Viana, Vasudevan Lakshminarayanan
Mappings φ:X↦Y preserving or representing the algebraic properties defined on sets X in Y are called homomorphisms from X into Y. There are, for example, group homomorphisms, ring homomorphisms, and algebra homomorphisms. The automorphisms of a structure X, as introduced in Chapter 1, are its bijective homomorphisms preserving the structures of interest. Bijective homomorphisms φ:X↦Y are called isomorphisms. We write X≃Y to indicate that the structures X and Y are algebraically the same, or isomorphic.
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A function f:X→Y $ f:X\rightarrow Y $ is surjective or onto Y if f(X)=Y $ f(X) = Y $ , and injective, or one-to-one (1-1), if x1≠x2⇒f(x1)≠f(x2) $ x_1 \not = x_2 \Rightarrow f(x_1) \not = f(x_2) $ . A surjection (injection) is a map that is surjective (injective). A mapping that is both surjective and injective is said to be bijective and is called a bijection or a one-to-one correspondence.
Preliminaries
Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020
Aliakbar Montazer Haghighi, Indika Wickramasinghe
In general, one may ask how many natural numbers are there? In other words, how far can one count? The answer, intuitively, is “the natural numbers are endless”. More formally, the “number” of natural numbers is infinite (cardinality), and this cardinality, denoted by ω, has the set of natural numbers, ℕ. If a set has the same number of elements as (i.e., if it is bijective with) the set of natural numbers, we say the set is denumerable or countably infinite, being also assigned cardinality ω. Also, a set is called countable if it is either finite or denumerable. If a set is not countable, it is uncountable with assigned cardinality one larger than ω.
LDPC Codes Based on Rational Functions
Published in IETE Journal of Research, 2021
Mohammad Gholami, Akram Nassaj
, where is the transpose of and is the inverse function of f.For two bijections and , we have in which is the composition of and ., where is the n times composition of f.
On m-Polar Interval-valued Fuzzy Graph and its Application
Published in Fuzzy Information and Engineering, 2020
Sanchari Bera, Madhumangal Pal
Letofandofbe two-PIVFG. An isomorphismis a bijective mappingsatisfying the following conditions,, , and, , and for each .
Traces of ternary relations
Published in International Journal of General Systems, 2018
Lemnaouar Zedam, Omar Barkat, Bernard De Baets
A permutation of a set X is a bijection from X to itself. For a set , we write instead of . The possible permutations of are listed as follows: