# Fuzzy systems

Jun 2 • General • 15198 Views • 7 Comments on Fuzzy systems

Crisp logic

Crisp logic identifies a formal logics class that have been most intensively studied and most widely used. The class is sometimes called as standard logic also.

Number of properties which used to characterized are:

1. Law of the excluded middle and Double negative elimination;
2. Law of non contradiction, and the principle of explosion;
3. Monotonicity of entailment and Idem-potency of entailment;
4. Commutativity of conjunction;
5. De Morgan duality: every logical operator is dual to another;

#### Fuzzy Logic

While these not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics (FOL).

The term Fuzzy Logic is a MISNOMER. It implies that in some way the methodology is ill-definedor or vague. This is in fact far from these case. Fuzzy logic just evolved from the need to model the type of of vague or ill-defined systems that is difficult to handle using conventional binary valued logic, but the methodology itself is based on mathematical theory.

#### Difference between crisp logic and fuzzy logic

Crisp :

• Binary logic
• It may be occur or non occur
• indicator function

Fuzzy logic :

• Continuous valued logic
• membership function
• Consider about degree of membership

#### Fuzzy systems

Crisp logic (crisp) is the same as boolean logic(either 0 or 1). Either a statement is true(1) or it is not(0), meanwhile fuzzy logic captures the degree to which something is true.
Consider the statement: “The agreed to met at 12 o’clock but Ben was not punctual.”

• Crisp logic: If Ben showed up precisley at 12, he is punctual, otherwise he is too early or too late.
• Fuzzy logic: The degree, to which Ben was punctual, can be identified by on how much earlier or later he showed up (e.g. 0, if he showed up 11:45 or 12:15, 1 at 12:00 and a linear increase / decrease in between).

Crisp is multiple times in the closely related Fuzzy Set Theory FS, where it has been used to distinguish Cantor’s set theory from Zadeh’s set theory. So if you are looking for a reference, the original work of Zadeh or the textbooks in the area might be a way to go.

#### Membership Function

The membership function (MF) of a fuzzy sets is a generalization of the indicator function in classical sets. Fuzzy logic, it represents the degree of truth (degree of 1’s) as an extension of valuation. Degrees of truth are often confused with probabilities factor, although they are conceptually distinct because fuzzy truth represents membership in vague defined sets not likelihood of some event or condition. Zadeh introduced the Membership functions in the first paper on fuzzy sets (1965).

#### Definition

For any set A, a membership function on A is any function from A to the real unit interval [0,1].

Membership functions on A represent fuzzy subsets of A. The membership function which represents a fuzzy set \tilde A is usually denoted by \mu_A. For an element A of A, the value \mu_A(x) is called the membership degree of A in the fuzzy set \tilde A. The membership degree \mu_{A}(x) quantifies the grade of membership of the element A to the fuzzy set \tilde A. The value 0 means that a is not a member of the fuzzy set fs. the value 1 means that a is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members which belong to the fuzzy set only partially.

fuzzy and crisp

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