OE-11.4 Classical & Fuzzy Sets
You can contrast classical with fuzzy sets, apply fuzzy operations, properties and relations, and construct membership functions.
Fuzzy logic replaces the yes/no of classical set membership with a degree between zero and one, which is how you make 'the water is warm' computable. The crucial distinction, and the one most often muddled, is that fuzziness is vagueness rather than probability - a 0.7 membership in 'tall' is not a 70 percent chance of being tall. Choosing membership functions is design rather than derivation; triangular and trapezoidal shapes dominate because they are cheap and good enough.
Work through these
Introduction to classical sets: properties, operations and relations
Ordinary set theory, revised because the next item modifies exactly these rules. Knowing which properties survive and which do not is the content.
Fuzzy sets and the concept of membership
The central idea of the second half of the course: membership as a degree rather than a yes or no. It is a small change with large consequences.
Uncertainty and vagueness versus probability
The distinction people get wrong most often. Vagueness is about where a boundary is; probability is about not knowing which side of a sharp boundary you are on.
Fuzzy set operations and properties
The operations, and which of the classical properties still hold. The law of the excluded middle failing is the one to remember.
Fuzzy relations
Relations between fuzzy sets, which is how rules connect inputs to outputs. It is the machinery the inference system in the next topic runs on.
Cardinalities of fuzzy sets
How big a fuzzy set is, when membership is partial. It is a small technical item needed by several of the defuzzification methods below.
Membership functions and their types
The shapes membership functions take, and how one is chosen. In practice the choice is made by hand from knowledge of the problem, which is worth being told plainly.
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