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...r-power set for matrix calculus . . . 51 3.2.1 Order based on the enumeration of isotone Boolean functions . . . 51 3.2.2 Ordering based on the DSm cardinality . . . 52 3.2.3 Ordering based on the intrinsic informational content . . . 56 3.3 Conclusion . . . . . 59 3.4 References . . . . . 60 4 Combination of beliefs on hybrid DSm models 61 4.1 Introduction . . . ....

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...hout this book S(n) will denote the semigroup of mappings of any set X with cardinality of X equal to n. Order of S(n) is denoted by ο (S(n)) or |S(n... ...e the term lattice for both concepts. |L| = o(L) denotes the order (that is cardinality) of the lattice L. Example 1.2.3: 5 4 L be a lattice give... ...reader to verify. Example 1.2.7: [- ∝, 1] is a chain lattice of infinite cardinality. Example 1.2.8: Take [0, 1] = L the two element set. L is ... ... o(S). If the number of elements in S is not finite we say S is of infinite cardinality. Now we give some examples of semirings. Example 2.1.1: ... ...Example 2.1.1: Let Z o = Z + ∪ {0}. (Z o , +, •) is a semiring of infinite cardinality and the characteristic Z o is 0. Further Z o is a commutat... ...+ ∪ {0}. (Q o , +, o) is also a commutative semiring with unit of infinite cardinality and characteristic Q o is 0. Example 2.1.3: Let M 2 ×2... ...teristic zero with unit element         1 0 0 1 and is of infinite cardinality. Example 2.1.4: Let S be the chain lattice given by the f... ...nd sup as binary operations on it. This is a commutative semiring of finite cardinality or order. This semiring has no characteristic associated wit... ...ified that this lattice is also a semiring which is commutative with finite cardinality and has no characteristic associated with it. DEFINITION 2...

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...G. An edge e is a cut edge of G, if {e} is an edge cut of G. An edge cut of cardinality K is called a K- edge cut of G. Example 2.3.2: Consider th... ...imal if G has no 41 matching M 1 strictly containing M. α 1 (G) is the cardinality of a maximum matching and β 1 (C) is the size of a minimum ... ... From a bipartite graph G with bipartition (X, Y) where X and Y are sets of cardinality p and q respectively say X = { υ 1 , υ 2 , …, υ p } and Y =... ...s of S; that is if u v ∈ E (G) then f (u) ≠ f (v). Thus χ(G) is the minimum cardinality of S for which there exists a proper vertex coloring of G by... ...nant vertices otherwise u can be equal to v. This χ N (G) is the minimum cardinality of S for which there exists a proper vertex colouring of G b...

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...e frame of discernment of hypotheses or universe of discourse by G. Shafer. The cardinality (number of elementary elements) of is denoted . The power ... ...he power set of which is the set of all subsets of is usually noted because its cardinality is exactly . Any element of is then a composite event (dis... ...ccessible which is unfortunately not possible in general as already argued. The cardinality of hyper power set is majored by when Card .T h e generati... ... of the Dedekind’s problem (for ) has not been found yet. We just know that the cardinality numbers of follow the integers of the Dedekind’s sequence ... ...h is not known and is still under investigations in the mathematical community. Cardinality numbers follow the Dedekind’s sequence (minus one), i.e. ... ...tropy of the source associated with bpa . This general definition introduces the cardinality of a general (irreductible) proposition which can be deriv... ...ntation (irreductible form). Examples Here are few examples of the value of the cardinality for some elementary and composite irreductible proposition...

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... for a biring R = R 1 ∪ R 2 to be Noetherian. THEOREM 1.5.49: If the cardinality of the image set of every bi-ideal of a biring R = R 1 ∪ R 2 ... ... subfield of F. 80 THEOREM 1.6.3: Let S be a subset of F such that S (Cardinality of S) ≥ 2. Then S is a subfield of F if and only if χ S , the... ...free sets over k of fuzzy singletons of V and every such set has the same cardinality. The following theorem gives condition for the existence of ... .../B has maximal fuzzy algebraically independent fuzzy subsets of F and the cardinality of each is unique. First we give some notational conventions... ...lgebraical. THEOREM 1.9.33: A/B has a fuzzy transcendence basis and the cardinality of a fuzzy transcendence basis is unique. In fact X is a fuzzy... ...fuzzy', and F(x) denotes the set of all fuzzy subsets of X. We denote the cardinality of o Aw h e r e A ∈ F(X ∗ ) by card A. 1 is the identity of X... ...[0, 1] ] is a vector space over R since the interval [0, 1] is of infinite cardinality we say R[x [0, 1] ] is a vector space with infinite basis. ... ...OREM 2.3.6: Let λ be a proper S-(∈,∈∨q)-fuzzy subgroup of G such that the cardinality of {λ(x) λ(x) < 0.5 } ≥ 2. Then λ can be expressed as the unio... ...f others then we say V is a finite dimensional S-semivector space and the cardinality of P is the dimension of V. We see in the case of S-semivec...

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...ed by the greatest solution and the set of minimal solutions. However, when the cardinality of either U of V is infinite, a few properties about the... ... exists at least one minimal solution x 0 = (x 0i ) ∈ ℵ ) (+ j . Moreover, the cardinality of {i ∈ | x oi > 0} is finite. Lemma 1.21.5: If x is ... ...3) Using the principle of parsimony, one might consider that the smaller the cardinality of D the more plausible it is. If M(D) can 129 be o... ...k manifestations in M + . This readily extends to subsets of causes of a given cardinality. In the general case, the weights where w σ(i) may lie i...

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...ed by the greatest solution and the set of minimal solutions. However, when the cardinality of either U of V is infinite, a few properties about the... ... exists at least one minimal solution x 0 = (x 0i ) ∈ ℵ ) (+ j . Moreover, the cardinality of {i ∈ | x oi > 0} is finite. Lemma 1.21.5: If x is ... ...3) Using the principle of parsimony, one might consider that the smaller the cardinality of D the more plausible it is. If M(D) can 129 be o... ...k manifestations in M + . This readily extends to subsets of causes of a given cardinality. In the general case, the weights where w σ(i) may lie i...

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...ned in a theory, this should be of infinite (and, even better, of aleph-) cardinality. Logicism: Frege's axioms for set theory, to derive... ...to absurd the reductio ad absurdum method? Continuum Hypothesis (that the cardinality of the continuum is the smallest non- denumerable cardinal) has...

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...subgroup H of S is such that H partitions S\{0} into equivalence class of equal cardinality; (S\{0} only if S contains {0}). PROBLEM 14: Find / ...

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...oughout this book S(n) will denote the set of all mapping of a set X with cardinality n to itself. i.e., X = {1, 2, … , n}; S(n) under the compositio...

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...e. iv. S- bistructure. v. perfect bistructure. 156. On a given set of cardinality n, how many S-bistructures can be defined? 157. Obtain any i...