Analisis sifat matematis pada fuzzy set

ANALISIS SIFAT MATEMATIS PADA FUZZY SET

Rahmat Purwoko, [email protected]

I. Pendahuluan

Tulisan ini membahas studi kasus untuk Fuzzy Set Membership Function dan

membuktikan apakah sifat-sifat matematis (dalam hal ini Set Theory) berlaku pada Fuzzy

Set tersebut. Pengujian dilakukan dengan cara membuat program untuk membuktikan

Set Thery dalam Scilab 5.5

No Set Theory Operation

1. Law of Contradiction A∩ A=∅

2. Law of The Excluded Middle A∪ A=X

3.Idempotency

A∩ A=A

A∪ A=A

4. Involution A=A

5.Commutativity

A∩B=B∩ A

A∪B=B∪A

6.Associativity

( A∪B )∪C=A∪ (B∪C )

( A∩B )∩C=A ∩ (B∩C )

7.Distibutivity

A∪ (B∩C )= (A∪B )∩ ( A∪C )

A∩ (B∪C )= (A∩B )∪ ( A∩C )

8.Absorption

A∪ ( A∩B )=A∪B

A∩ ( A∪B )=A∩B

9.Absorption of Complement

A∪ ( A∩B )=A∪B

A∩ ( A∪B )=A∩B

10.De Morgan’s Laws

A∪B=A∩B

A∩B=A∪B

Studi kasus yang diambil adalah masalah intensitas cahaya dalam ruangan dengan

satuan Luks. Dalam hal ini semesta pembicaraan (x) adalah 0 – 250. Pembentukan

membership function menggunakan triangular dengan parameter :

A = tri_mf(x,[-125,0,125]);

B = tri_mf(x,[0,125,250]);

C = tri_mf(x,[125,250,325]);

Hasil plot grafik dapat dilihat pada gambar berikut.

II. Analisis Sifat Matematis (Set Theory) pada Fuzzy Set

1. Law of Contradiction

A∩ A=∅

Analisis : Plot Grafik tidak sesuai antara ruas kiri dan ruas kanan persamaan.

2. Law of The Excluded Middle

A∪ A=X

2

Analisis : Plot Grafik tidak sesuai antara ruas kiri dan ruas kanan persamaan.

3. Idempotency

a. Persamaan 1 : A∩ A=A

b. Persamaan 2 : A∪ A=A

Analisis : Plot Grafik sesuai antara ruas kiri dan ruas kanan persamaan.

4. Involution

Persamaan : A=A

3


5. Commutativity

a. Persamaan 1 : A∪B=B∪A

b. Persamaan 2 :A∩B=B∩ A


6. Associativity

a. Persamaan 1: A∪ (B∪C )= (A∪B )∪C

4

b. Persamaan 2 :A∩ (B∩C )=(A ∩B )∩C


7. Distributivity

a. Persamaan 1 : A∪ (B∩C )= (A∪B )∩ ( A∪C )

5

b. Persamaan 2 :A∩ (B∪C )= (A∪B )∩ ( A∪C )


8. Absorption

a. Persamaan 1 : A∪ ( A∩B )=A

b. Persamaan 2 : A∩ ( A∪B )=A


9. Absorption of Complement

6

a. Persamaan 1 : A∪ ( A∩B )=A∪B

b. Persamaan 2 : A∩ ( A∪B )=A∩B


10. De Morgan’s Law

a. Persamaan 1 : A∪B=A∩B

b. Persamaan 2 : A∩B=A∪B

7


III. Summary

Hasil pengujian Set Theory dengan Plot grafik untuk Fuzzy Set diatas diperlihatkan dalam

tabel berikut :

No Set Theory Operation Result

1. Law of Contradiction A∩ A=∅ Tidak Sesuai

2. Law of The Excluded Middle A∪ A=X Tidak Sesuai

3.Idempotency

A∩ A=A

A∪ A=A

Sesuai

Sesuai

4. Involution A=A Sesuai

5.Commutativity

A∩B=B∩ A

A∪B=B∪A

Sesuai

Sesuai

6.Associativity

( A∪B )∪C=A∪ (B∪C )

( A∩B )∩C=A ∩ (B∩C )

Sesuai

Sesuai

7.Distibutivity

A∪ (B∩C )= (A∪B )∩ ( A∪C )

A∩ (B∪C )= (A∩B )∪ ( A∩C )

Sesuai

Sesuai

8.Absorption

A∪ ( A∩B )=A∪B

A∩ ( A∪B )=A∩B

Sesuai

Sesuai

9. Absorption of Complement A∪ ( A∩B )=A∪B Sesuai

8

A∩ ( A∪B )=A∩B Sesuai

10.De Morgan’s Laws

A∪B=A∩B

A∩B=A∪B

Sesuai

Sesuai

Sedangkan pengujian dengan cara membandingkan element matriks dari masing-masing

operasi Set Theory menghasilkan beberapa perbedaan, hal ini dikarenakan operasi yang

digunakan dalam orde 0 s/d 1. Sehingga terjadi pembulatan saat operasi complement (1-

A). Berikut log hasil pengujiannya :

-------------------------------------------------------------------------

Running Set Theory Analysis ...

Analysis Result :

------------------------------------------------------------------------

1. Law of Contradiction :

Plot tidak sesuai. Persamaan tidak terpenuhi

------------------------------------------------------------------------

2. Law of The Excluded Middle :

------------------------------------------------------------------------

Persamaan 1


------------------------------------------------------------------------

2. Law of The Excluded Middle :

------------------------------------------------------------------------

Persamaan 1


------------------------------------------------------------------------

3. Idempotency :

------------------------------------------------------------------------

Persamaan 1

Plot sesuai. Persamaan terpenuhi

Persamaan 2


------------------------------------------------------------------------

4. Involution :

------------------------------------------------------------------------


------------------------------------------------------------------------

5. Distributivity :

------------------------------------------------------------------------

Persamaan 1


Persamaan 2


9

------------------------------------------------------------------------

6. Associativity :

------------------------------------------------------------------------

Persamaan 1


Persamaan 2


------------------------------------------------------------------------

7. Distributivity :

------------------------------------------------------------------------

Persamaan 1


Persamaan 2


------------------------------------------------------------------------

8. Absorption :

------------------------------------------------------------------------

Persamaan 1


Persamaan 2


------------------------------------------------------------------------

9. Absorption of Complement :

------------------------------------------------------------------------

Persamaan 1


Persamaan 2


------------------------------------------------------------------------

10. Analysis De Morgans Laws :

------------------------------------------------------------------------

Persamaan 1


Persamaan 2


------------------------------------------------------------------------

IV. Source Code

A. tri_mf.sci

function [y] = tri_mf(x, parameter)

// TRI_MF Triangular membership function with three parameters.

// TRI_MF(x, [a, b, c]) returns a matrix y with the same size

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// as x; each element of y is a grade of membership.

// Jyh-Shing Roger Jang, 6-28-93.

// Ported to Scilab by Rahmat Purwoko, 2014

a = parameter(1); b = parameter(2); c = parameter(3);

if a > b,

error('Illegal parameters: a > b');

elseif b > c,

error('Illegal parameters: b > c');

elseif a > c,

error('Illegal parameters: a > c');

end

y = max(min((x-a)/(b-a), (c-x)/(c-b)), 0);

endfunction

B. Tugas1.sce

// Tugas I

// Analisa Sifat Matematis (Set Theory) pada Fuzzy Set

// Mata Kuliah : Kendali dan Sistem Cerdas

// Dosen : Prof. Dr. Ir. Bambang Riyanto Trilaksono, M.Sc

// Software : Scilab 5.5 (Open Source)

// Close all opened figures and clear workspaces

xdel(winsid());

clear;

clc;

//change or run it first

exec('D:\ITB 2014\SKC\tugas 1\source code\tri_mf.sci', -1)

// Universe

x = 0:250

// Fuzzy Set Generation using Triangular Membership Function

A = tri_mf(x,[-125,0,125]);

B = tri_mf(x,[0,125,250]);

C = tri_mf(x,[125,250,325]);

// 0. Fuzzy Set membership function

subplot(2,2,1);

plot(x, A, x, B, x, C);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];

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a.parent.background=4;

a.background=4;

legend(["A", "B", "C"]);

xtitle("Triangular ABC MF", "Ilummination Level", "Membership Function");

// ------------------------------------------------------------------------- //

// 1. Law of Contradiction

// ------------------------------------------------------------------------- //

figure

// Left

mf1a1 = min(A,1-A);

subplot(2,2,1)

plot(x,mf1a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$ A\cap \overline{A}$");

xtitle("$Law\ of\ Contradiction - A\cap \overline{A}$", "Ilummination Level",

"Membership Function");

// Right

subplot(2,2,2)

a = gca();

a.axes_visible="on";

a.data_bounds=[0,250,-0.1,1.2,];

a.grid =[1 1];

a.box="on";

xtitle("$Law\ of\ Contradiction - \emptyset$", "Ilummination Level", "Membership

Function");

// ------------------------------------------------------------------------- //

// 2. Law of The Excluded Middle

// ------------------------------------------------------------------------- //

figure

// Left

mf2a1 = max(A,1-A);

subplot(2,2,1);

plot(x,mf2a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


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a.background=4;

legend("$A\cup \overline{A}$");

xtitle("Law of The Excluded Middle", "Ilummination Level", "Membership Function");

// Right

mf2a2 = x;

subplot(2,2,2);

plot(x,mf2a2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$X$");

xtitle("Law of The Excluded Middle", "Ilummination Level", "Membership Function");

// ------------------------------------------------------------------------- //

// 3. Idempotency

// ------------------------------------------------------------------------- //

figure

// Left

mf3a1 = min(A,A);

subplot(2,2,1);

plot(x,mf3a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cap A$");

xtitle("Idempotency(1)", "Ilummination Level", "Membership Function");

//Right

mf3a2 = A;

subplot(2,2,2);

plot(x,mf3a2);

a = gca();

a.grid =[1 1];

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a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A$");


// Persamaan 2

// Left

mf3b1 = max(A,A);

subplot(2,2,3);

plot(x,mf3b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cup A$");


// Persamaan 2

// Right

mf3b2 = A;

subplot(2,2,4);

plot(x,mf3b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A$");


// ------------------------------------------------------------------------- //

// 4. Involution

// ------------------------------------------------------------------------- //

figure

// Left

mf4a1 = 1-(1-A);

subplot(2,2,1);

plot(x,mf4a1);

a = gca();

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a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$\overline{\overline{A}}$");

xtitle("Involution", "Ilummination Level", "Membership Function");

// Right

mf4a2 = A;

subplot(2,2,2);

plot(x,mf4a2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A$");

xtitle("Involution", "Ilummination Level", "Membership Function");

// ------------------------------------------------------------------------- //

// 5. Commutativity

// ------------------------------------------------------------------------- //

figure

// Persamaan 1

// Left

mf5a1 = max(A,B);

subplot(2,2,1);

plot(x,mf5a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cup B$");

xtitle("Commutativity (1)", "Ilummination Level", "Membership Function");

// Persamaan 1

// Right

mf5a2 = max(B,A);

subplot(2,2,2);

plot(x,mf5a2);

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a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$B\cup A$");


// Persamaan 2

// Left

mf5b1 = min(A,B);

subplot(2,2,3);

plot(x,mf5b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cap B$");


// Persamaan 2

// Right

mf5b2 = min(B,A);

subplot(2,2,4);

plot(x,mf5b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$B\cap A$");


// ------------------------------------------------------------------------- //

// 6. Associativity

// ------------------------------------------------------------------------- //

figure

// Persamaan 1

// Left

mf6a1 = max(A,max(B,C));

subplot(2,2,1);

plot(x,mf6a1);

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a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cup (B\cup C)$");

xtitle("Associativity (1)", "Ilummination Level", "Membership Function");

// Persamaan 1

// Right

mf6a2 = max(max(A,B),C);

subplot(2,2,2);

plot(x,mf6a2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$(A\cup B)\cup C$");


// persamaan 2

// Left

mf6b1 = min(A,min(B,C));

subplot(2,2,3);

plot(x,mf6b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cap (B\cap C)$");


// Persamaan 2

// Right

mf6b2 = min(min(A,B),C);

subplot(2,2,4);

plot(x,mf6b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


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a.background=4;

legend("$(A\cap B)\cap C$");


// ------------------------------------------------------------------------- //

// 7. Distributivity

// ------------------------------------------------------------------------- //

figure

// Persamaan 1

// Left

mf7a1 = max(A,min(B,C));

subplot(2,2,1);

plot(x,mf7a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cup (B\cap C)$");

xtitle("Distributivity (1)", "Ilummination Level", "Membership Function");

// Persamaan 1

// Right

mf7a2 = min(max(A,B),max(A,C));

subplot(2,2,2);

plot(x,mf7a2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$(A\cup B)\cap (A\cup C)$");


// persamaan 2

// Left

mf7b1 = min(A,max(B,C));

subplot(2,2,3);

plot(x,mf7b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


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a.background=4;

legend("$A\cap (B\cup C)$");


// Persamaan 2

// Right

mf7b2 = max(min(A,B),min(A,C));

subplot(2,2,4);

plot(x,mf7b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$(A\cap B)\cup (A\cap C)$");


// ------------------------------------------------------------------------- //

// 8. Absorption

// ------------------------------------------------------------------------- //

figure

// Persamaan 1

// Left

mf8a1 = max(A,min(A,B));

subplot(2,2,1);

plot(x,mf8a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cup (A\cap B)$");

xtitle("Absorption (1)", "Ilummination Level", "Membership Function");

// Persamaan 1

// Right

mf8a2 = A;

subplot(2,2,2);

plot(x,mf8a2);

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a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A$");


// persamaan 2

// Left

mf8b1 = min(A,max(A,B));

subplot(2,2,3);

plot(x,mf8b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cap (A\cup B)$");


// Persamaan 2

// Right

mf8b2 = A;

subplot(2,2,4);

plot(x,mf8b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A$");


// ------------------------------------------------------------------------- //

// 9. Absorption of Complement

// ------------------------------------------------------------------------- //

figure

// Persamaan 1

// Left

mf9a1 = max(A,min(1-A,B));

subplot(2,2,1);

plot(x,mf9a1);

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a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cup (\overline{A}\cap B)$");

xtitle("Absorption of Complement (1)", "Ilummination Level", "Membership

Function");

// Persamaan 1

// Right

mf9a2 = max(A,B);

subplot(2,2,2);

plot(x,mf9a2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$: A\cup B$");


Function");

// persamaan 2

// Left

mf9b1 = min(A,max(1-A,B));

subplot(2,2,3);

plot(x,mf9b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$A\cap (A\cup B)$");


Function");

// Persamaan 2

// Right

mf9b2 = min(A,B);

subplot(2,2,4);

plot(x,mf9b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];

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a.background=4;

legend("$: A\cap B$");


Function");

// ------------------------------------------------------------------------- //

// 10. De Morgan's Laws

// ------------------------------------------------------------------------- //

figure

// Persamaan 1

// Left

mf10a1 = 1-max(A,B);

subplot(2,2,1);

plot(x,mf10a1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$\overline{A\cup B$");

xtitle("De Morgan`s Laws (1)", "Ilummination Level", "Membership Function");

// Persamaan 1

// Right

mf10a2 = min(1-A,1-B);

subplot(2,2,2);

plot(x,mf10a2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$\overline{A}\cap \overline{B}$");


// Persamaan 2

// Left

mf10b1 = 1-min(A,B);

subplot(2,2,3);

plot(x,mf10b1);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


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a.background=4;

legend("$\overline{A\cap B$");


// Persamaan 2

// Right

mf10b2 = max(1-A,1-B);

subplot(2,2,4);

plot(x,mf10b2);

a = gca();

a.grid =[1 1];

a.data_bounds=[0,250,-0.1,1.2,];


a.background=4;

legend("$\overline{A}\cup \overline{B}$");


// ------------------------------------------------------------------------- //

// Simple Analisis Generator

// Generate True or False based on equivalencies matrix

// ------------------------------------------------------------------------- //

disp "------------------------------------------------------------------------"

disp "------------------------------------------------------------------------"

disp "Running Set Theory Analysis ..."

disp "Analysis Result : "

disp "------------------------------------------------------------------------"

disp "1. Law of Contradiction :"

if mf1a1 == [] then

disp "Plot sesuai. Persamaan terpenuhi"

else

disp "Plot tidak sesuai. Persamaan tidak terpenuhi"

end

disp "------------------------------------------------------------------------"

disp "2. Law of The Excluded Middle :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1 "

if mf2a1 == mf2a2 then


else

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end

disp "------------------------------------------------------------------------"

disp "2. Law of The Excluded Middle :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else


end

disp "------------------------------------------------------------------------"

disp "3. Idempotency :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else


end

disp "Persamaan 2"

if mf3b1 == mf3b2 then


else


end

disp "------------------------------------------------------------------------"

disp "4. Involution :"

disp "------------------------------------------------------------------------"



else


end

disp "------------------------------------------------------------------------"

disp "5. Distributivity :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else


end

disp "Persamaan 2"


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else


end

disp "------------------------------------------------------------------------"

disp "6. Associativity :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else


end

disp "Persamaan 2"



else


end

disp "------------------------------------------------------------------------"

disp "7. Distributivity :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else


end

disp "Persamaan 2"



else


end

disp "------------------------------------------------------------------------"

disp "8. Absorption :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else

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end

disp "Persamaan 2"



else


end

disp "------------------------------------------------------------------------"

disp "9. Absorption of Complement :"

disp "------------------------------------------------------------------------"

disp "Persamaan 1"



else


end

disp "Persamaan 2"



else


end

disp "------------------------------------------------------------------------"

disp "10. Analysis De Morgans Laws :"

disp "------------------------------------------------------------------------"



else


end

disp "------------------------------------------------------------------------"

// ------------------------------------------------------------------------- //

// Save Figgure to PNG file

// ------------------------------------------------------------------------- //

for i = 0:10

xs2png(i, sprintf("tugas1_%03d.png",i))

sleep(100);

end

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Analisis sifat matematis pada fuzzy set

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Transcript of Analisis sifat matematis pada fuzzy set