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 Book Id: WPLBN0002096589 Subjects: Non Fiction, Education, Algebra Collections: Math, Literature, Education, Algebra, Authors Community, Mathematics ► Abstract Full Text Search Details... of a set A with itself; A × A. Note that if the set A is a finite set having n elements, then the set A × A is also a finite set, but has n 2 elem... ...nts, then the set A × A is also a finite set, but has n 2 elements. The set of elements (a, a) in A × A is called the 8 diagonal of A × A. A s... ...bout subsets of A × A we can now speak about a binary relation (one between two elements of A) on A itself, by defining b to be related to a if (a, ... ...chapters of this book. The example deals with nothing but mapping of a set of n elements to itself. Example 1.2.1: Let (1, 2, 3) be the set S. Le... ...he set S. Let S(3) denote the set of all mappings of S to itself. The number of elements in S(3) is 27 = 3 3 . 10 Example 1.2.2: Let S = (1, 2... ...theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. sch...
 Book Id: WPLBN0002097100 ► Abstract Full Text Search Details....6: Let G(I) be a neutrosophic vector space over the field K. The number of elements in the neutrosophic basis is called the neutrosophic dimension ... ...g neutrosophic vector space over the neutrosophic field K(I). The number of elements in the strong neutrosophic basis is called the strong neutrosop... ...tinguished from the others), with point of degree at most 4 and finally the chemical problem of trees in which every point has degree 1 or 4. Jorda... ...e that Jordan did so without having any suspicion of its bearing on modern chemical doctrine. The most famous problem in graph theory and perhaps i... ...nite non empty set V = V (G) of p points. (vertex, node, junction O-simplex elements) together with a prescribed set X of q unordered pairs of disti... ...Each pair x = {u, v} of points in X is a line (edge, arc, branch, 1-simplex elements) of G and x is said to join u and v. We write x = uv and say t... ...graph is a complete graph. Thus in a tournament for every pair of distinct elements u and v either (u, v) or (v, u) but not both is an arc of D. To... ...ry, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars...
 Book Id: WPLBN0002097044 Collections: Math, Literature, Algebra, Authors Community, Mathematics ► Abstract Full Text Search Details...(B 1 + B 2 ). Thus two bimatrices are added by adding the corresponding elements only when compatibility of usual matrix addition exists. No... ...y square bimatrix then ( ) 2 mm mm II × × = . A bimatrix A B all of whose elements are zero is called a null or zero bimatrix and it is denoted by... ...This is a m th order square bimatrix. This will not have 2m 2 arbitrary elements since 11 ij ji aa = and 22 ij ji aa = (where A 1 = 1 ij (a ... ...nd A 2 = 2 ij (a )) both below and above the main diagonal. The number of elements above the main diagonal of A B = A 1 ∪ A 2 is (m 2 – m). Th... ... above the main diagonal of A B = A 1 ∪ A 2 is (m 2 – m). The diagonal elements are also arbitrary. Thus the total number of arbitrary elements... ...ry, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars...
 Book Id: WPLBN0002097045 ► Abstract Full Text Search Details...∪ (B 1 + B 2 ). Thus two bimatrices are added by adding the corresponding elements only when compatibility of usual matrix addition exists. ... ...dence maps that associates with each element of E i (G) an unordered pair of elements of V i (G) i = 1, 2. It is to be noted that E 1 (G 1 ) ... ...simple. A bigraph G = G 1 ∪ G 2 is bisimple if |G| = n that is number of elements in G 1 and G 2 without repetition i.e. o (G 1 ∪ G 2 ) = o(... ...ion between the health hazards the agricultural labourer suffers 94 due to chemical pollution. Let us taken the 11 attributes (P 1 , P 2 ,…, P 11... ... – Ulcer / skin ailments in legs and hands P 3 – Manuring the fields with chemical fertilizers P 4 – Vomiting P 5 – Mouth and stomach ... ...e at a fixed bipoint as the bihidden pattern. Thus manuring the field with of chemical fertilizers makes all the coordinates on except P 4 which im... ...ey may not suffer the symptom of vomiting because of manuring the field with chemical fertilizers. Now we study the effect of the attribute Head... ...ed by them due to spray of pesticides insecticides and manuring the plants by chemical fertilizers. At the first stage we take arbitrary attributes ... ...concepts. We further assume no intermediate relation exists within the domain elements or node and the range spaces elements. The number of elements...
 Book Id: WPLBN0002097649 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details... loop-rings and introduction to Smarandache loop-rings 20 2.3 Smarandache elements in loop rings 30 2.4 Smarandache substructures in loop rings 40 ... ... Groupoid rings and Smarandache Groupoid rings 61 3.3 Smarandache special elements in Groupoid rings 69 3.4 Smarandache substructures in Groupoid ri... ...pace consists of the following: 1. a field K of scalars. 2. a set V of elements called vectors. 3. a rule or operation called vector addition whi... ...ly it is a vector quantity, only to mention that it is different from the elements of the field which are termed as scalars. 8 DEFINITION 1.1.2... ...s finite dimensional if there exists a finite set of linearly independent elements (v 1 , v 2 , …, v n ) in V which generate V, otherwise it is infi... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. scholars a...
 Book Id: WPLBN0002097647 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...ngs and its properties 2.1 Definition and examples 21 2.2 Special elements in rings 24 2.3 Substructures of a ring 26 2.4 Homomorph... .... and S. D.C.C. 67 3.8 Some special types of rings 68 3.9 Special elements in S-rings 71 3.10 Special properties about S-rings 78 ... ... 115 4.2 Smarandache rings of level II 119 4.3 Some new Smarandache elements and their properties 121 4.4 New Smarandache substructures and th... ... a -1 a = aa -1 = e. A group, which contains only a finite number of elements, is called a finite group, otherwise it is termed as an infinite ... ...s an infinite group. By the order of a finite group we mean the number of elements in the group. 8 It may happen that a group G consists enti... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. scholars ...
 Book Id: WPLBN0002097643 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...n P by declaring a ∗ b = c if σ (a, b) = c. DEFINITION: A non empty set of elements G is said to form a groupoid if in G is defined a binary oper... ...all a ∈ G. We call the order of the groupoid G to be the number of distinct elements in it denoted by o(G) or |G|. If the number of elements in G ... ...order of this groupoid is 5. Example 1.2.2: Let (S, ∗) be a groupoid with 3 elements given by the following table: ∗ x 1 x 2 x 3 x 1 x 1 x... ... commutative semigroup. DEFINITION: Let S be a semigroup, if the number of elements in a semigroup is finite we say the semigroup S is of finite ... ...= {0, 1}} is a semigroup under matrix multiplication. PROBLEM 9: How many elements are there in S 3×3 given in example 1.3.6.? PROBLEM 10: Ca... ...eory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. schola...
 Book Id: WPLBN0002097651 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...3 2 1 Clearly µ is a fuzzy subgroup of a group G and o(µ ) = number of elements of the set {x ∈ G | µ (x) = µ (e)} = number of elements of the se... ... [0, min{µ 1 (e 1 ), µ 2 (e 2 )}] where e 1 and e 2 denote the identity elements of the groups G 1 and G 2 respectively. Then t 1 t 2 t 1 2 G G... ...R µ , the ring of fuzzy cosets of µ in R is free from non-zero nilpotent elements. THEOREM 1.4.23: Let µ be any fuzzy ideal of a ring R such that... ... that Im µ = {t, s} with t > s. If the ring R µ has no non-zero nilpotent elements, then the fuzzy ideal µ is fuzzy semiprime. THEOREM [99]: A ri... ... If P : R → L is a prime fuzzy ideal and P P (0) ≠ 0, then P (R) has two elements. P is properly fuzzy if and only if P(R) has three elements. We s... ...Rings and Ideals, Addison-Wesley, Reading, MA, 1970. 30. CHAUVIN, Remi, Chemical Algebra I: Fuzzy subgroups, J. Math. Chemistry, 16, 245-256 (1994... ..., fuzzy groups, and applications of fuzzy theory to the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars a...
 Book Id: WPLBN0002097043 Subjects: Non Fiction, Algebra, Smarandache Collections ► Abstract Full Text Search Details...by default of notation just say V a vector space over the field F and call elements of V as vectors only as matter of convenience for the vectors in... ... dimensional vector space) of k if every S- basis has only finite number of elements in it. It is interesting to note that if L is a finite dimension... ...We say the order of the bigroup G = G 1 ∪ G 2 is finite if the number of elements in them is finite; otherwise we say the bigroup G to be of infin... ...ace V = V 1 ∪ V 2 to be of dimension m + n. Thus there exists only m + n elements which are linearly independent and has the capacity to generate... ...0) if v ∈ V 2 then v = (0, 0, …, 0, y 1 , y 2 , …, y n ). We never add elements of V 1 and V 2 . We keep them separately as no operation may be... ...y, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars ...
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