*Fuzzy IF-THEN Rules*

Firstly, recall the fuzzy logic operations and, or, not, implication, and Equivalence also:

a ∧ b, a ∧ b, a¯, a ⇒ b, a ⇔ b,

And their evaluations on a fuzzy sets, A, with the membership function μ_{A}(.):

μ_{A} (a ∧ b) : = min {μ_{A} (a), μ_{A} (b)} = μ_{A} (a) ∧ μ_{A} (b);

μ_{A} (a ∨ b) : = max {μ_{A} (a), μ_{A} (b)} = μ_{A} (a) ∨ μ_{A} (b);

μ_{A } (a ) = μ_{A} (a) = 1 - μ_{A} (a)

μ_{A} (a ⇒ b) = μ_{A} (a) ⇒ μ_{A} (b) = min {1, 1 + μ_{A} (a) - μ_{A} (b)}

μ_{A} (a ⇔ b) = μ_{A} (a) ⇔ μ_{A} (b) = 1 - | μ_{A} (a) - μ_{A} (b) |

Recall the fuzzy relations also among elements of two fuzzy sets A and B, on which a membership function μ_{A×B}(a, b) is defined, along with a ∈ A and b ∈ B. This is clear that one can certainly consider the above fuzzy logic computations as several special fuzzy relations, along with A = B and μ_{A×A} = μ_{A}.

The implication relation a ⇒ b such can be interpreted, in linguistic terms, like

*"IF a is true THEN b is true."*

For fuzzy logic performed upon a fuzzy set A, there is a membership function μ_{A} explaining the truth values of a ∈ A and b ∈ A. In this case, a more complete linguistic statement would be as

"(IF a ∈ A is true with a truth value μ_{A} (a) THEN b ∈ A is true with a truth value μ_{A }(b)) has the truth value μ_{A} (a ⇒ b) = min {1, 1 + μA (b) - μA (a)}."

In the above, both a and b belongs to the similar fuzzy subset A and share the similar membership function μ_{A}. One has a non trivial fuzzy relation, which can be fairly complicated, if they belong to various fuzzy sets A and B with various membership functions μ_{A} and μ_{B}. However, In most cases, the implication relation a ⇒ b, performed on fuzzy sets A and B, here a ∈ A and b ∈ B, is simply defined in linguistic terms like

"(IF a ∈ A is true with a truth value μ_{A} (a) THEN b ∈ A is true with a truth vale μ_{B} (b)) has the truth value μ_{A×B} (a ⇒ b) = min {1, 1 + μ_{B} (b) - μ_{A} (a)}."

As all such statements have a standard format and their meaning is inside clear context, this is usual to write them in the following simple form as:

**"IF a is A THEN b is B."**

A fuzzy logic implication statement of this form is usually called a fuzzy IF- THEN rule. To be more usual, let A_{1}, A_{2}, . . . , A_{n} and B be the fuzzy subsets along with membership functions μ_{A1}, μ_{A2}, . . . , μ_{An}, and μ_{B}, respectively.

*Definition of fuzzy logic.1*

Commonly fuzzy IF- THEN rule has the form

"IF a1 is A1 AND...AND an is An THEN b is B."

By utilizing the fuzzy logic AND operation, such rule is implemented by the specified evaluation formula as:

μ_{AI} (a1 ) ∧ ... ∧ (an) ⇒ μB (b) ,

Here

μ_{Ai }= (a_{1} ) ∧ μ_{Aj}(a_{j}) = min {μ_{A} (a_{i} ), μ_{A} (a_{j})}

And, thus,

μ_{l}(a_{1}) ∧ ... ∧ μ_{A} (a_{n}) = min{μ_{A} (a_{1} ), ... , μ_{A} (a_{n})}

About this general or common fuzzy IF-THEN rule and its evaluation, a few issues have to be clarified as:

- There is no fuzzy logic OR operation in a general or common fuzzy IF-THEN rule.

What should one do if a fuzzy logic implication statement includes the OR operation?

- There is no fuzzy logic NOT operation in a general or common fuzzy IF-THEN rule.