Chapter 5

a.

b. The two inequalities (5.1) and (5.2) are consequences of the definition of the δ-cuts of the fuzzy relative frequencies. If the support of a fuzzy observation has non-empty intersection with two classes Kj and Kj + 1, then this observation is counted for n, δ(Kj) and for n, δ(Kj + 1). This observation is counted twice in the sum n,δ(Kj) + n, δ(Kj + 1). If taking the union Kj ∪ Kj + 1 for n, δ(Kj ∪ Kj + 1) the observation is counted only once. From this inequality (5.1) is obtained. For the lower boundaries n, δ(Kj) and n, δ(Kj + 1), and n, δ(Kj ∪Kj + 1) it is possible that the ...

Start Free Trial

No credit card required