By Professor John N. Mordeson, Associate Professor Premchand S. Nair (auth.)
In the mid-1960's I had the excitement of attending a conversation through Lotfi Zadeh at which he provided a few of his uncomplicated (and on the time, contemporary) paintings on fuzzy units. Lotfi's algebra of fuzzy subsets of a suite struck me as really nice; actually, as a graduate scholar within the mid-1950's, I had steered related principles approximately continuous-truth-valued propositional calculus (inffor "and", sup for "or") to my consultant, yet he did not opt for it (and actually, stressed it with the principles of likelihood theory), so i stopped up writing a thesis in a extra traditional sector of arithmetic (differential algebra). I specifically loved Lotfi's dialogue of fuzzy convexity; I take note speaking to him approximately attainable methods of extending this paintings, yet i did not pursue this on the time. i've got somewhere else informed the tale of ways, while I observed C. L. Chang's 1968 paper on fuzzy topological areas, i used to be impelled to attempt my hand at fuzzi fying algebra. This resulted in my 1971 paper "Fuzzy groups", which grew to become the place to begin of a complete literature on fuzzy algebraic constructions. In 1974 King-Sun Fu invited me to talk at a U. S. -Japan seminar on Fuzzy units and their functions, which was once to be held that summer time in Berkeley.
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Then Me = He . 17 For any fuzzy graph G and any T > 0, the elements of G of cohesiveness at least T form a fuzzy graph whose components are T-edge components of G . 18 If G' is an T-edge component of the fuzzy graph G for some T > 0, then G' = He for some element e of G . • 36 2. •• , Cm), is a slicing of G if each member G, ffi a cut-"", (A;,A,) of { ~\ ;Q: G j for i = 1 for 2 ::; i ::; m A member of the slicing will also be termed a cut of the slicing. A slicing of G, which is minimal in the sense that there is no subpartition which is a slicing of G, is called a minimal slicing of G.
We have three different cases to consider: (1) U,V E VI \ V2, (2) u E VI \ V2,V E Vl nV2 and (3) U,V E Vl nV2 . (i) Let U,V E VI \ V2. Then (El U E2)(UV) (AI U A 2)(u) 1\ (AI U A 2)(v). = El(uv) 'S Al(U) 1\ Al(V) = (ii) Let u E VI \ V2 and v E VI n V2. Then (El U E2)(UV) 'S (AI U A2)(U) 1\ (AI (v) V A 2(v)) = (AI U A 2)(u) 1\ (AI U A 2)(v). (iii) Let u, v E VI n V2 . Then (El U E2)(UV) 'S (Al(U) V A2(U)) 1\ (AI (v) V A2(V)) = (AI U A2)(U) 1\ (AI U A2)(V), Similarly, ifuv E X 2 \X l , then (E l UE2)(uv) 'S (AlUA2)(U)I\(AlUA2)(V), Suppose that uv E Xl n X 2.
1 Let G be a fuzzy graph. Let 0 <::: f <::: 1. A vertex v is said to be E-reachable fTOm another vertex u if there exists a positive integer k such that Rk(u,v) 2' E. The reachability matrix ofG. denoted by Mk=, -is the matrix of the fuzzy graph (A, R=). fi>ned as follows: M =(u, v) = lif R( u, v) 2' E and M = (u, v) = 0, otherwise. k A k The following algorithm can determine the reachability between any pair of vertices in a fuzzy graph G. 1. Determination of M ROO 1. Let Ri = (ail, ... , ain) denote the ith row.