## Fuzzy Logic and Sets

This theory lets us handle and process information in a similar way as the human brain does.

We communicate and coordinate actions with data like ".you are too young to do that." How much does "too" refer to, what's "young"?

With fuzzy sets, we may define sub-sets in a fashion that any element may be part of them in different degrees.

With fuzzy rules, it's possible to compute relationships between fuzzy variables and produce fuzzy outputs.

And guess what, from these fuzzy output values; we may build boolean and continuous quantities, like a switch status or an amount of money.

"In the future, western people will employ Fuzzy logic concepts as they have done with Boolean logic since Aristotle (eastern people already do it in some way) " (My humble opinion).

## Welcome to our tutorial on Fuzzy logic and Sets.

Fuzzy math involves in general three operations:

Fuzzification: Translation from real-world values to Fuzzy values.:

Rule evaluation: Computing rule strengths based on rules and inputs.:

Defuzzyfication: Translate results back to the real world values.

## FUZZIFICATION

It makes the translation from real-world values to Fuzzy world values using membership functions.

The membership functions in Fig.1, translate a speed= 55 into fuzzy values (Degree of membership),

SLOW=0.25, MEDIUM=0.75, and FAST=0.

## RULE EVALUATION:

You may have a rule like this: If SPEED=SLOW and HOME=FAR then GAS=INCREASES

Suppose SLOW=0.25 and FAR=0.82.
The rule strength will be 0.25 (The minimum value of the antecedents) and the fuzzy variable INCREASE would be also 0.25.

Consider now another rule: If SPEED=MEDIUM and HIGHER=SECURE then GAS=INCREASE

Let be in this case, MEDIUM=0.75 and SECURE=0.5. Now the rule strength will be 0.5 (The minimum value of the antecedents) and the fuzzy variable INCREASE would be also 0.5.

So, we have two rules involving the fuzzy variable INCREASE. The "Fuzzy OR" of the two rules will be 0.5 (The maximum value between the two proposed values).

## DEFUZZYFICATION:

After computing the fuzzy rules and evaluate the fuzzy variables, we will need to translate these results back to the real world. We need now a membership function for each output variable like in Fig. 2.

Let the fuzzy variables be: DECREASE=0.2, SUSTAIN=0.8, and INCREASE=0.5

Each membership function will be clipped to the value of the correspondent fuzzy variable as shown in fig.3.

A new output membership function is built, taking for each point in the horizontal axis, the maximum value between the three membership values. The result is shown in Fig. 4.

To complete the Defuzzyfication process, all we have to do now is found an equilibrium point. One way to do this is with "Center of Gravity method" (COG).

The above expression may be evaluated numerically, computing a few discrete terms instead of an integral, giving in this case approximately,

... time for a fun TEST (Please give me a few days to recover my JS script and update it, thank you!).

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