Fuzzy IF-THEN Rules Assignment Help

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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 μA(.):

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

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

μA  (a ) = μA  (a) = 1 - μA  (a)

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

μA  (a ⇔ b) = μA  (a) ⇔ μA  (b) = 1 - | μA (a) - μA  (b) |

Recall the fuzzy relations also among elements of two fuzzy sets A and B, on which a membership function μA×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 μA×A = μA.

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 μA 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 μA (a) THEN b ∈ A is true with a truth value μA (b)) has the truth value μA (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 μA. 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 μA and μB. 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 μA (a) THEN b ∈ A is true with a truth vale μB (b)) has the truth value μA×B (a ⇒ b) = min {1, 1 + μB (b) - μA (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 A1, A2, . . . , An and B be the fuzzy subsets along with membership functions μA1, μA2, . . . , μAn, and μB, 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:

 μAI (a1 ) ∧ ... ∧ (an) ⇒ μB (b) ,


 μAi = (a1 ) ∧ μAj(aj) = min {μA (ai ), μA (aj)}

And, thus,

μl(a1) ∧ ... ∧ μA (an) = min{μA (a1 ), ... , μA (an)}

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.
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