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# Fuzzy Logic Rule Base Assignment Help

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Fuzzy Rules And Their Operations - Fuzzy Logic Rule Base
**Fuzzy Logic Rule Base**

Consider questions rose above. Firstly, consider, for illustration, the following fuzzy

**IF-THEN rule containing an OR operation:**

IF a_{1} is A_{1} AND a_{2} is A_{2} OR a_{3} is A_{3} AND a_{4} is A_{4} THEN b is B." By convention, it is understood in logic as

"(IF a_{1} is A_{1} AND a_{2} is A_{2}) OR (a_{3} is A_{3} AND a_{4} is A_{4}) THEN b is B."

With this understanding and convention, this is clear that this statement is Equivalent to the combination of the given two fuzzy IF-THEN rules as:

"IF a_{1} is A_{1} AND a_{2} is A_{2} THEN b is B." "IF a_{3} is A_{3} AND a_{4} is A_{4} THEN b is B."

Thus, the fuzzy logic OR operation is not necessary to employ: this may shorten a statement of a fuzzy IF-THEN rule; however it increases the format complexity of the rules.

Secondly, consider the fuzzy logic NOT operation. For a negative statement like "IF a is not A," one can always interpret it by a positive one "IF a¯ is A" or "IF a is A" where A means "not A" in logic theory and also "complement of A" in set theory. Moreover, the statement "a¯ is A" or "a is A" can be evaluated by

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

**Illustration 1**

Specified a fuzzy logic implication statement

"IF a_{1} is A_{1} AND a_{2} is not A_{2} OR a_{3} is not A_{3 }THEN b is B,"

How one can rewrite this as a set of Equivalent general fuzzy IF-THEN rules in the unified form?

**Solution**

One may firstly drop the fuzzy logic OR operation by rewriting the specified statement as

"IF a_{1} is A1 AND a_{2} is not A_{2} THEN b is B,"

"IF a_{3} is not A_{3} THEN b is B,"

One may then drop the fuzzy logic NOT operation by rewriting them like

"IF a_{1} is A_{1} AND a¯_{2} is not A_{2} THEN b is B,"

"IF a¯_{3 } is A_{3} THEN b is B,"

Finally, these two general fuzzy IF-THEN rules can be evaluated as follows

μ_{A1}(a_{1}) ∧ μ_{A2} (a¯_{2} ) ⇒ μ_{B}(b)

μ_{A3} (a¯_{3}) ⇒ μ_{B}(b)

Hence, one only needs two general fuzzy IF-THEN rules, (1) and (2), and three membership ship values μ_{A1}(a_{1}), μ_{A2}(a_{2}), μ_{A3}(a_{3}) to infer the conclusion "b is B," namely, b ∈ B, along with the truth values μ_{B}(b) calculated from the three specified membership values.

Every other fuzzy logic operations can be simply expressed and defined only by the AND and OR operations. They can be evaluated through min and the max operations as given below:

μA(a1 ) ∧ μA (a2) : min {μ1(a1), μA(a2)};

Among

μ_{A}(a¯) ∧ μA (a_{2}) : max {μ_{1} (a_{1} ), μ_{A}

μ_{A} (a ) = μ_{A} (a) = 1 - μ_{A}(a) (a_{2})};

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) |

Consequently, each finite combinations of these fuzzy logic operations can be expressed only by the AND and OR operations, hence in any finite fuzzy logic inference statement

IF ... THEN ...,

While the condition part "IF ..." can be expressed only by the AND and OR operations.

Consequently, a finite fuzzy logic implication can usually be described by a set of general or common IF-THEN rules containing merely the fuzzy logic AND operation, in the given generic form as:

(a) "IF a_{11} is A_{11} AND . . . AND a_{1n} is A_{1n} THEN b_{1} is B_{1}."

(b) "IF a_{21} is A_{21} AND . . . AND a_{2n} is A_{2n} THEN b_{2} is B_{2}."

..........

(c) "IF a_{m1} is A_{m1} AND . . . AND a_{mn} is A_{mn} THEN b_{2} is B_{2}."

This family of general or common fuzzy IF_THEN rules is mostly called a fuzzy logic rule base.

This is remarked that the number of components in all rule needs not to be the similar. If n = 2 however a rule has one component in the situation part, say as,

"IF a_{11} is A_{11} THEN b_{1} is B_{1},"

One cam formally rewrites this as

""IF a_{11} is A_{11} AND a_{12} is I_{12} THEN b_{1} is B_{1}." Here I_{12} is a fuzzy subset along with μ_{12} (a) = 1 for all a ∈ I_{12}

Now, one actually inserts an "always true" or redundant condition into the "IF ... AND ..." part to fill in the gap of the statement. In a common discussion of the subject the format of a fuzzy logic rule base can be remained simple.

This is clear that such general forms of a fuzzy logic rule base includes the non-fuzzy logic rule base involve the non fuzzy case and the unconditional case with only "b is B" as special cases. Furthermore, this common fuzzy logic rule base with merely the fuzzy logic AND operation in the situation part too covers too unusual fuzzy logic implication statements, as like the one shown in the later illustration.

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**Fuzzy Logic Rule Base**

**IF-THEN rule containing an OR operation:**_{1}is A

_{1}AND a

_{2}is A

_{2}OR a

_{3}is A

_{3}AND a

_{4}is A

_{4}THEN b is B." By convention, it is understood in logic as

_{1}is A

_{1}AND a

_{2}is A

_{2}) OR (a

_{3}is A

_{3}AND a

_{4}is A

_{4}) THEN b is B."

_{1}is A

_{1}AND a

_{2}is A

_{2}THEN b is B." "IF a

_{3}is A

_{3}AND a

_{4}is A

_{4}THEN b is B."

_{A}(a¯) = μ

_{A}(a) = 1 - μ

_{A}(a)

**Illustration 1**_{1}is A

_{1}AND a

_{2}is not A

_{2}OR a

_{3}is not A

_{3 }THEN b is B,"

**Solution**_{1}is A1 AND a

_{2}is not A

_{2}THEN b is B,"

_{3}is not A

_{3}THEN b is B,"

_{1}is A

_{1}AND a¯

_{2}is not A

_{2}THEN b is B,"

_{3 }is A

_{3}THEN b is B,"

_{A1}(a

_{1}) ∧ μ

_{A2}(a¯

_{2}) ⇒ μ

_{B}(b)

_{A3}(a¯

_{3}) ⇒ μ

_{B}(b)

_{A1}(a

_{1}), μ

_{A2}(a

_{2}), μ

_{A3}(a

_{3}) to infer the conclusion "b is B," namely, b ∈ B, along with the truth values μ

_{B}(b) calculated from the three specified membership values.

_{A}(a¯) ∧ μA (a

_{2}) : max {μ

_{1}(a

_{1}), μ

_{A}

_{A}(a ) = μ

_{A}(a) = 1 - μ

_{A}(a) (a

_{2})};

_{A}(a ⇒ b) = μ

_{A}(a) ⇒ μ

_{A}(b) = min {1, 1 + μ

_{A}(a) - μ

_{A}(b)}

_{A}(a ⇔ b) = μ

_{A}(a) ⇔ μ

_{A}(b) = 1 -| μ

_{A}(a) - μ

_{A}(b) |

_{11}is A

_{11}AND . . . AND a

_{1n}is A

_{1n}THEN b

_{1}is B

_{1}."

_{21}is A

_{21}AND . . . AND a

_{2n}is A

_{2n}THEN b

_{2}is B

_{2}."

_{m1}is A

_{m1}AND . . . AND a

_{mn}is A

_{mn}THEN b

_{2}is B

_{2}."

_{11}is A

_{11}THEN b

_{1}is B

_{1},"

_{11}is A

_{11}AND a

_{12}is I

_{12}THEN b

_{1}is B

_{1}." Here I

_{12}is a fuzzy subset along with μ

_{12}(a) = 1 for all a ∈ I

_{12}