By W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
This booklet offers the elemental notions of fuzzy matrix idea and its functions to basic versions. The method is non-traditional that allows you to allure many scholars to take advantage of this technique of their examine. the conventional strategy of mathematicians has conditioned scholars of sociology in this type of demeanour that they averse to utilizing mathematical instruments. Six uncomplicated sorts of fuzzy versions that utilize fuzzy matrices are given. those types are exact simply because they're time established and will also be used for statistical information. the bushy Cognitive Maps version provides the hidden development. Fuzzy Relational Maps version not just offers the inter-relations among units of disjoint attributes. The Bidirectional Associative stories version analyzes information counting on the time-period whereas the bushy Associative stories version can provide the gradation of significance of every characteristic. eventually, the bushy Relational Equation version is in a position to giving an answer in the direction of the anticipated resolution. the entire equipment are illustrated via problematic examples of specific social difficulties.
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6 a single set. The related is follows. 3⎥ . 16 The graph representation by a single diagram. Study in this direction using this graph will be carried out for any appropriate models which is under investigation. 3: A binary relation R (X, X) that is reflexive and symmetric is usually called a compatibility relation or tolerance relation. 3: Consider a fuzzy relation R (X, X) defined on X = {x1, x2, …, x7} by the following membership matrix. 1 0 ⎥ . 0 0 0 0 1 ⎥⎦ Graph of the compatibility relation or the compatibility relation graph.
Min product. 7 ⎥ ⎢ ⎥ ⎢ 1 ⎥ Y = ⎢ ⎥. Y using the max min product. 7 . Y is a singleton fuzzy matrix. Now using the same max. 4 Here also we see XY ≠ YX. 6 ⎥ ⎥. 5 ⎦⎥ we have a fuzzy square 1⎤ 0 1 ⎥⎥ 1 0⎥ ⎥ 0 1⎥ −1 0 ⎥ ⎥ 0 1⎥ 1 0 ⎥⎦ 1 Let Y = [1 1 1 1 1 1 1], ⎡ −1 ⎢1 ⎢ ⎢0 ⎢ AY = ⎢ −1 ⎢0 ⎢ ⎢1 ⎢1 ⎣ 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 46 1⎤ 0 1 ⎥⎥ 1 0⎥ ⎥ 0 1⎥ −1 0 ⎥ ⎥ 0 1⎥ 1 0 ⎥⎦ 1 ⎡1⎤ ⎡ 3 ⎤ ⎢1⎥ ⎢ 4 ⎥ ⎢⎥ ⎢ ⎥ ⎢1⎥ ⎢ 3 ⎥ ⎢⎥ ⎢ ⎥ ⎢1⎥ = ⎢ 1 ⎥ . ⎢1⎥ ⎢ 1 ⎥ ⎢⎥ ⎢ ⎥ ⎢1⎥ ⎢ 4 ⎥ ⎢1⎥ ⎢ 4 ⎥ ⎣⎦ ⎣ ⎦ We see that multiplication of A by the column unit matrix Y gives us the row sums of the fuzzy matrix like if we multiply X = [1 1 1 1 1 1 1] the row unit matrix with the fuzzy matrix A will give the column sums.
Such a function is called a membership function and the set defined by 34 it a fuzzy set. As we are not going to work with fuzzy sets or its related graphs we advice the interested reader to refer to [106]. These books supply a lot of information about fuzzy sets. We have just defined fuzzy sets and the need of its in our study. Now we proceed on to define various types of fuzzy matrices without going deep into their structure. Only what is needed for the models are given, for more properties refer [106].