Editing Fuzzy logic (section)

====Fuzzification====

Fuzzification is the process of assigning the numerical input of a system to fuzzy sets with some degree of membership. This degree of membership may be anywhere within the interval [0,1]. If it is 0 then the value does not belong to the given fuzzy set, and if it is 1 then the value completely belongs within the fuzzy set. Any value between 0 and 1 represents the degree of uncertainty that the value belongs in the set. These fuzzy sets are typically described by words, and so by assigning the system input to fuzzy sets, we can reason with it in a linguistically natural manner.

For example, in the image below, the meanings of the expressions ''cold'', ''warm'', and ''hot'' are represented by functions mapping a temperature scale. A point on that scale has three "truth values"—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot"; i.e. this temperature has zero membership in the fuzzy set "hot". The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold". Therefore, this temperature has 0.2 membership in the fuzzy set "warm" and 0.8 membership in the fuzzy set "cold". The degree of membership assigned for each fuzzy set is the result of fuzzification.

[[Image:Fuzzy logic temperature en.svg|thumb|center|upright=1.5|Fuzzy logic temperature]]

[[Fuzzy set]]s are often defined as triangle or trapezoid-shaped curves, as each value will have a slope where the value is increasing, a peak where the value is equal to 1 (which can have a length of 0 or greater) and a slope where the value is decreasing.<ref>{{Cite journal |last1=Xiao |first1=Zhi |last2=Xia |first2=Sisi |last3=Gong |first3=Ke |last4=Li |first4=Dan |date=2012-12-01 |title=The trapezoidal fuzzy soft set and its application in MCDM |journal=Applied Mathematical Modelling |language=en |volume=36 |issue=12 |pages=5846–5847 |doi=10.1016/j.apm.2012.01.036 |issn=0307-904X|doi-access=free }}</ref> They can also be defined using a [[sigmoid function]].<ref>{{cite web|last1=Wierman|first1=Mark J.|title=An Introduction to the Mathematics of Uncertainty: including Set Theory, Logic, Probability, Fuzzy Sets, Rough Sets, and Evidence Theory|url=https://www.creighton.edu/fileadmin/user/CCAS/programs/fuzzy_math/docs/MOU.pdf|publisher=Creighton University|access-date=16 July 2016|url-status=live|archive-url=https://web.archive.org/web/20120730155249/https://www.creighton.edu/fileadmin/user/CCAS/programs/fuzzy_math/docs/MOU.pdf|archive-date=30 July 2012}}</ref> One common case is the [[Logistic function|standard logistic function]] defined as

: <math> S(x) = \frac{1}{1 + e^{-x}} </math>,

which has the following symmetry property

: <math display="block"> S(x) + S(-x) = 1.</math>

From this it follows that

<math display="block"> (S(x) + S(-x)) \cdot (S(y) + S(-y)) \cdot (S(z) + S(-z)) = 1 </math>