## Fuzzy Sets Explained A fuzzy set is a set whose elements have degrees of membership. Unlike crisp sets, where an element either belongs to a set or doesn't (membership is either 0 or 1), fuzzy sets allow for partial membership. This is represented by a membership function that assigns a value between 0 and 1 (inclusive) to each element in the universe of discourse. A value of 1 indicates full membership, 0 indicates no membership, and values between 0 and 1 represent varying degrees of membership. **Example:** Consider the fuzzy set "Tall People". There's no single height that definitively separates "tall" from "not tall". We can define this fuzzy set using a membership function. Let's say the universe of discourse is heights in centimeters, ranging from 150cm to 200cm. A possible membership function could be: * **µ(x) = 0** if x ≤ 160cm (definitely not tall) * **µ(x) = (x - 160) / 20** if 160cm < x < 180cm (gradually increasing membership) * **µ(x) = 1** if x ≥ 180cm (definitely tall) Here: * µ(x) represents the membership function. * x represents the height in centimeters. This function assigns a membership degree to each height. For example: * µ(150cm) = 0 * µ(170cm) = 0.5 * µ(180cm) = 1 * µ(190cm) = 1 This shows that a person of 170cm is considered "half-tall" according to this particular fuzzy set definition. The exact shape and parameters of the membership function depend on the context and the subjective interpretation of "tall". ## Complement of a Fuzzy Set The complement of a fuzzy set A, denoted by A', is obtained by subtracting the membership grade of each element from 1. Formally: **µ_A'(x) = 1 - µ_A(x)** Where: * µ_A'(x) is the membership function of the complement of A. * µ_A(x) is the membership function of A. **Example (continued):** Let's find the complement of the "Tall People" fuzzy set defined above. The membership function of the complement, "Not Tall People", would be: * **µ_A'(x) = 1** if x ≤ 160cm * **µ_A'(x) = 1 - (x - 160) / 20** if 160cm < x < 180cm * **µ_A'(x) = 0** if x ≥ 180cm This shows how the membership grades are inverted: a person previously considered "half-tall" (170cm) is now "half-not tall". The complement captures the opposite concept within the fuzzy logic framework.