# Fuzzy rule

**Fuzzy rules** are used within fuzzy logic systems to infer an output based on input variables. Modus ponens and modus tollens are the most important rules of inference.[1] A modus ponens rule is in the form

- Premise:
*x is A* - Implication:
**IF**x is A**THEN**y is B - Consequent:
*y is B*

In crisp logic, the premise *x is A* can only be true or false. However, in a fuzzy rule, the premise *x is A* and the consequent *y is B* can be true to a degree, instead of entirely true or entirely false.[2] This is achieved by representing the linguistic variables *A* and *B* using fuzzy sets.[2] In a fuzzy rule, modus ponens is extended to *generalised modus ponens:.[2]*

- Premise:
*x is A** - Implication:
**IF**x is A**THEN**y is B - Consequent:
*y is B**

The key difference is that the premise *x is A* can be only partially true. As a result, the consequent *y is B* is also partially true. Truth is represented as a real number between 0 and 1, where 0 is false and 1 is true.

## Comparison between Boolean and fuzzy logic rules

As an example, consider a rule used to control a three-speed fan. A binary IF-THEN statement may be

**IF***temperature**30***THEN***fan speed is 3*

The disadvantage of this rule is that it uses a strict temperature as a threshold, but the user may want the fan to still function at this speed when temperature = 29.9. A fuzzy IF-THEN statement may be

**IF***temperature is hot***THEN***fan speed is fast*

where *hot* and *fast* are described using fuzzy sets.

## Fuzzy rule connectors

Rules can connect multiple variables through fuzzy set operations using t-norms and t-conorms.

**T-norms** are used as an *AND* connector.[3][4][5] For example,

**IF***temperature is hot***AND**humidity is high**THEN***fan speed is fast*

A degree of truth is assigned to *temperature is hot* and to *humidity is high.* The result of a t-norm operation on these two degrees is used as the degree of truth that *fan speed is fast*.

**T-conorms** are used as an *OR* connector.[5] For example,

**IF***temperature is hot***OR**humidity is high**THEN***fan speed is fast*

The result of a t-conorm operation on these two degrees is used as the degree of truth that *fan speed is fast*.

The complement of a fuzzy set is used as a negator.[5] For example,

**IF***temperature is***NOT**hot**THEN***fan speed is slow*

The fuzzy set *not hot* is the complement of *hot.* The degree of truth assigned to *temperature is not hot* is used as the degree of truth that *fan speed is slow*.

T-conorms are less commonly used as rules can be represented by *AND* and *OR* connectors exclusively.

## See also

## References

- B., Enderton, Herbert (2001).
*A mathematical introduction to logic*(2nd ed.). San Diego, Calif.: Academic Press. ISBN 978-0122384523. OCLC 45830890. - 1938-, Mendel, Jerry M. (2001).
*Uncertain rule-based fuzzy logic systems : introduction and new directions*. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 978-0130409690. OCLC 45314121. - Martin Larsen, P. (1980). "Industrial applications of fuzzy logic control".
*International Journal of Man-Machine Studies*.**12**(1): 3–10. doi:10.1016/s0020-7373(80)80050-2. ISSN 0020-7373. - Mamdani, E.H. (1974). "Application of fuzzy algorithms for control of simple dynamic plant".
*Proceedings of the Institution of Electrical Engineers*.**121**(12): 1585. doi:10.1049/piee.1974.0328. ISSN 0020-3270. - H.-J., Zimmermann (1991).
*Fuzzy Set Theory - and Its Applications*(Second, revised ed.). Dordrecht: Springer Netherlands. ISBN 9789401579490. OCLC 851369348.