# fuzzy logic

Discussion in 'Math' started by vead, Jul 11, 2014.

Nov 24, 2011
621
8
2. ### Papabravo Expert

Feb 24, 2006
10,338
1,850
There are three mapping functions called "slow", "average", and "fast"

"slow" maps the value 19 to 0.6, by definition
"average" maps the value 19 to 0.4, again by definition
"fast" maps the value 19 to to 0, again, you guessed it by definition

That is how it's done.

3. ### tshuck Well-Known Member

Oct 18, 2012
3,531
675
Look at the graph under "Membership Function", it shows that the value 19 belongs to the three membership functions at the point it crosses the plot of each of the membership functions.

Nov 24, 2011
621
8
I don't understand how does μf become 0.4 , 0.6 how to determine this value

5. ### tshuck Well-Known Member

Oct 18, 2012
3,531
675
In the example given, there are 3 membership functions: fast, average, and slow.

Each of these is plotted on the same graph, with the following piece-wise definitions:
$\mu_{slow}(x) = \left\{
\begin{array}{lr}
1\,:\,x \in [10,18)
y\,=\,-0.6x +12 &\,:\,x \in [18,20]
0\,:\,otherwise
\end{array}
\right.$

$\mu_{average}(x) = \left\{
\begin{array}{lr}
y\,=\,0.6x -11 &\,:\,x \in [18.\overline{3},20]
y\,=\,-0.5x +11 &\,:\,x \in (20,22]
0\,:\,otherwise
\end{array}
\right.$

$\mu_{fast}(x) = \left\{
\begin{array}{lr}
y\,=\,0.5x -10 &\,:\,x \in [20,22]
1\,:\,x \in [22,25]
0\,:\,otherwise
\end{array}
\right.$

Then, the x-value is determined for each membership function, so, for 19, we have:
$\mu_{slow}(19)\,=\,-0.6(19) +12 = 0.6$
$\mu_{average}(19) =\,0.6(19) -11 = 0.4$
$\mu_{fast}(19) = 0$

Nov 24, 2011
621
8
I am still confused look at this simple example

Let S is a set of educated people where by educated we mean minimum graduate. In this case, the universal set is all types of people with various level of education which we classify as:

Level no.(x) Qualification
0 No education
1 Elementary School

2 High School
3 Two yr. College degree
4 Bachelor’s degree
5 Master’s degree
6 Doctoral degree

But if we define the set S as Fuzzy set then we have to assign membership grade (or degree of truth which is ‘member of the set’) to every element.

1 1 1
6 5 4 3 2 1 0

how to determine membership grade for uneducated people μ(x) ?

Last edited: Jul 13, 2014
7. ### tshuck Well-Known Member

Oct 18, 2012
3,531
675
By definition (provided you are assuming this set captures all possible values), the fuzzy set for uneducated people is 1 minus the sum of all others memberships.

Those characterizations are pretty crisp values and won't benefit much from fuzzy logic. If you want to use fuzzy logic, you could have something like the following sets: poorly educated, moderately educated, and well educated, and define ranges where these values fit, for instance:
$\mu_{poorly \, educated}(x) = \left\{
\begin{array}{lr}
1\,:\,x \in [0,1)
y\,=\,-0.333x + 1.333&\,:\,x \in [1,4)
0\,:\,x \in [4,6]
\end{array}
\right.$

$\mu_{moderately \, educated}(x) = \left\{
\begin{array}{lr}
0\,:\,x \in [0,2)
y\,=\,\frac{2}{3}x -1 &\,:\,x \in [2,3)
y\,=\,-\frac{1}{2}x + 2.5&\,:\,x \in [3,5)
0\,:\,x \in (5,6]
\end{array}
\right.$

$\mu_{well \, educated}(x) = \left\{
\begin{array}{lr}
0\,:\,x \in [0,4)
y\,=\,0.5x -1.5 &\,:\,x \in [4,5)
1\,:\,x [5,6]
\end{array}
\right.$

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