3Set Operations of Fuzzy Sets

We shall study the intersection and union of fuzzy sets. Let upper A overtilde and upper B overTilde be two fuzzy sets in double-struck upper R Superscript m with membership functions xi Subscript upper A overtilde and xi Subscript upper B overTilde, respectively. The usual intersection and union of upper A overtilde and upper B overTilde by referring to Zadeh [162] are defined using the min and max aggregation functions as follows

(3.1)xi Subscript upper A overtilde intersection upper B overTilde Baseline left-parenthesis x right-parenthesis equals min left-brace right-brace comma of xi xi upper A overtilde left-parenthesis right-parenthesis x comma of xi xi upper B tilde left-parenthesis right-parenthesis x and xi Subscript upper A overtilde union upper B overTilde Baseline left-parenthesis x right-parenthesis equals max left-brace right-brace comma of xi xi upper A overtilde left-parenthesis right-parenthesis x comma of xi xi upper B tilde left-parenthesis right-parenthesis x period

For more detailed properties, refer to Mizumoto and Tanaka [81], Dubois and Prade [28], and Klir and Yuan [60].

We can extend the and aggregation functions to the the t‐norm and s‐norm (t‐conorm), ...

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