# Fuzzy Logic and Education: Teaching the Basics of Fuzzy Logic through an Example (by Way of Cycling)

## Abstract

**:**

## 1. Introduction. Fuzzy logic and education

## 2. Boolean logic, probability and fuzzy logic

_{P}. The curve determines the degree to which a given object satisfies certain property or not. Graded properties are represented by fuzzy sets. A fuzzy set over a universe X is defined by its generalized membership function, usually denoted by µ

_{P}(x), representing the compatibility to attribute the property P to a given element x or, in other interpretation, the possibility to soundly attribute P to x. In this latter sense, as Zadeh noted in [16], it is said that a vague predicate is characterized by its possibility distribution, i.e., by the specification of the degree to which each element of the universe is compatible with the meaning of the predicate. By way of example, in the universe U=[0, 10], the vague predicate ‘small’ can be represented by the following possibility distribution:

_{F}, µ

_{S}).

_{F˄S}(x)=min(f

_{F}(x), f

_{S}(x)). Thus, the new sentence will be represented by this new function (Figure 4):

## 3. Teaching the basics of fuzzy logic through an example (by way of cycling)

- P1: On a bicycle, with a small freewheel and a large chainring, I go fast.
- P2: I selected a very small freewheel and a very large chainring.

- C: I go very fast.

- A representation of the vague lexicon included in the premises (in bold): ‘On a bicycle, with a
**small**freewheel and a**large**chainring, I go**fast**and ‘I selected a**very small**freewheel and a**very large**chainring’. - A method for inferring a conclusion. As a conclusion follows from imprecise premises, it must be approximate, providing information according to what the premises do (‘I go really fast’)

_{A}, representing the degree to which each element of the universe of discourse is compatible with what the vague predicate means. But one may ask why to propose this representation and not others, such as the following:

_{1}-a

_{2}=n(a

_{2})-n(a

_{1}) for all a

_{1}, a

_{2}∈[0, 1], n(a)=1-a is the only negation. If that condition is not met, there are other negation operators. For example, if we represent the meaning of the predicate ‘small’ in U = [0, 1] by the following membership function:

Name | Notation | Definition |
---|---|---|

bounded | T_{0} | Max(0,a+b-1) |

S_{0} | Min(1, a+b) | |

product | T_{1} | a×b |

S_{1} | a+b-a×b | |

min/max | T_{2} | min(a,b) |

S_{2} | max(a,b) |

_{L}(a, b)=min(1, 1-a+b); if S is max, we obtain the Kleene-Dienes conditional: I

_{K}(a, b)=max(1-a, b).

- Compositional inference
- Inference by analogy or compatibility
- Interpolative inference

If X is A then Y is C |

A’ is γ compatible with (or similar to) A |

_{1}and C

_{2}to approximately the same extent that A’ is between A

_{1}and A

_{2}.

Perception or fact | X is A* |

#### 3.2 Numerical Models for representing the example

- P1: On a bicycle, with a small freewheel and a large chainring, I go fast.
- P2: I selected a very small freewheel and a very large chainring.

μ_{SF}(x)= {12/0.9; 16/0.6; 20/0.4; 24/0.1} |

μ_{LC}(y)= {33/0.1; 52/0.9} |

μ_{GF}(z)= {3/0.1; 6/0.5; 9/0.9} |

_{SF}(x) × μ

_{LC}(y)=μ

_{BR}(x,y) and its values are reflected in the following table:

μ_{BR}(x,y) | 12 | 16 | 20 | 24 |
---|---|---|---|---|

33 | 0.09 | 0.06 | 0.04 | 0.01 |

52 | 0.81 | 0.0.54 | 0.36 | 0.09 |

_{BR→GF}(w). First, we analyze the rule we intend to model interpreting the meaning and the dependence or independence of their predicates, choosing the most appropriate conditional operator. In this case, ‘If I select a big ratio, I go fast’, can be interpreted as:

- I select a big ratio and I go fast.
- Or I do not select a big ratio or I go fast
- Or I do not select a big ratio, or I do and I go fast.
- Or I do not select a big ratio or I go at a speed proportional to the ratio selected.

μ_{BR→GF}(w) | 12-33 | 12-52 | 16-33 | 16-52 | 20-33 | 20-52 | 24-33 | 24-52 |
---|---|---|---|---|---|---|---|---|

3 | 0.919 | 0.0.271 | 0.9406 | 0.514 | 0.964 | 0.676 | 0.991 | 0.919 |

6 | 0.955 | 0.595 | 0.97 | 0.74 | 0.98 | 0.82 | 0.995 | 0.955 |

9 | 0.9181 | 0.919 | 0.994 | 0.946 | 0.996 | 0.964 | 0.999 | 0.991 |

_{P}(u)=x, μ

_{very_P}(u)=x

^{2}. Taking into account this definition:

μ_{very_BR}(x,y) | 12 | 16 | 20 | 24 |
---|---|---|---|---|

33 | 0.0081 | 0.0036 | 0.0016 | 0.0001 |

52 | 0.6561 | 0.2916 | 0.1295 | 0.0081 |

_{A}*(w):

μ_{GF*}(w)= {3/0; 6/0.251; 9/0.5751} |

_{GF}for these same elements. The value for 9 is the most divergent (it should be around 0.81 and, in fact, it is 0.5751). How can be expressed this answer? This matter is called, in the realm of fuzzy logic, ‘linguistic approximation’ and there are different techniques for solving it. Generally speaking, we can check that the obtained set μ

_{GF}

_{*}is more similar to one that ideally represent ‘go very fast’ than to any other, and that is precisely the conclusion that a rational agent draws from the premises. The conclusion is approximate because the reasoning is so too; there is nothing special about it.

#### 3.3 Linguistic Models

- If the freewheel is small and the chainring is large, the progress made is remarkable
- If the freewheel is medium and the chainring is medium, the progress made is moderate
- If the freewheel is large and the chainring is small, little progress is made.

## 4. Conclusions

## Acknowledgements

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Sobrino, A. Fuzzy Logic and Education: Teaching the Basics of Fuzzy Logic through an Example (by Way of Cycling). *Educ. Sci.* **2013**, *3*, 75-97.
https://doi.org/10.3390/educsci3020075

**AMA Style**

Sobrino A. Fuzzy Logic and Education: Teaching the Basics of Fuzzy Logic through an Example (by Way of Cycling). *Education Sciences*. 2013; 3(2):75-97.
https://doi.org/10.3390/educsci3020075

**Chicago/Turabian Style**

Sobrino, Alejandro. 2013. "Fuzzy Logic and Education: Teaching the Basics of Fuzzy Logic through an Example (by Way of Cycling)" *Education Sciences* 3, no. 2: 75-97.
https://doi.org/10.3390/educsci3020075