By Jan Jantzen(auth.)

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Suppose R is an mby- p matrix and S is a p-by-n matrix. Then the inner ∪ − ∩ product (read ‘cup-cap product’) is an m-by-n matrix T = (ti j ) whose i jth entry is obtained by combining the ith row of R with the jth column of S, such that p ti j = ri1 ∩ s1 j ∪ ri2 ∩ s2 j ∪ . . 4) k=1 The notation looks unwieldy, but the operation is essentially a matter of combining rows with columns successively, as the next example shows. 11). 6 which agrees with the previous result. With our previous definitions of the set operations, composition is specifically called the max–min composition (Ross 2010).

8). It is reasonable to demand that fuzzy connectives agree with their classical counterparts defined in the truth-domain 0, 1 . That is, in terms of truth-tables, the values in the four corners of the fuzzy Cayley table should agree with the Cayley table for the classical connective. Indeed, the four corners of ‘fuzzy or’ check with the table for two-valued ‘or’. Now derive the truth-table for ‘nand’ from ‘or’ by the definition (¬ p) ∨ (¬q) by negating the variables. Moving further, ‘and’ is the negation of the entries in the truth-table for ‘nand’.

The shaded areas are: union A ∪ B (a, d), intersection A ∩ B (b, e), and complement A ∪ B (c, f). m) For example, {1, 2, 3} ∩ {1, 3, 4} = {1, 3} The (classical) complement of a set X , symbolized by X and read ‘the complement of X ’, is X ≡ {x | x ∈ / X} That is, the set of those members of the universe that are not members (∈) / of X . 4, clearly illustrate the set operations. A Venn diagram is a picture showing sets as circles or closed curves, to illustrate relations between sets and objects.

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