Different between crisp set and fuzzy set?

Answer 1

In short, for a crisp set (subset) elements of the set definitely do belong to the set, while in a fuzzy set (subset) elements of the set have a degree of membership in the set. To make things clearer:

Suppose we have a reference set X={x_1, ...} and a subset Y={y_1, ...} of X. If Y represents a crisp subset of X, then for all x_n belonging to X, x_n either belongs or Y or does not belong to Y. We can write this by assigning a function C which takes each member of X to 1 iff it belongs to Y, and 0 iff it does not belong to Y. E. G. Suppose we have the set {1, 2, 3, 4, 5}. For the crisp subset {1, 2, 4} we could write this in terms of a function C which takes 1 to 1, 2 to 1, 3 to 0, 4 to 1, and 5 to 0, or we can write {(1, 1), (2, 1), (3, 0), (4, 1), (5, 1)}.

For a fuzzy subset F of a reference set X the elements of F may belong to F to a degree in between 0 and 1 (as well as may belong to F to degree 0 or 1). We can write this by assigning a function M which takes each member of X to a number in the interval of real numbers from 0 to 1, [0, 1] to represent its degree of membership. Here "larger" numbers represent a greater degree of membership in the fuzzy subset F. For example, for the reference set {1, 2, 3, 4, 5} we could have a function M which takes 1 to .4, 2 to 1, 3 to .6, 4 to .2, and 5 to 0, or {(1, .4), (2, 1), (3, .6), (4, .2), (5, 0)}, with 3 having a greater degree of membership in F than 4 does, since .6>.2.