FUNDAMENTALS OF OPERATIONS ON FUZZY SETS
There have been quite a few questions on operations on fuzzy sets in every exam of NET(in PAPER III). So I have explained the theory behind them and how to solve it. Hope it is useful.
1. How to denote a fuzzy set?
IF X is the universe of discourse and x is a particular element of X, then a fuzzy set A defined on X may be written as a collection of ordered pairs:
A={(x,µA(A))}, x belongs to X
The pair (x, µA(A)) is called a singleton.
In crisp sets, a singleton is simply the element x by itself.
In fuzzysets, a singleton is composed of two terms: x and µA(x).
A singleton is also written as µA(x)/x. That is by putting the membership function first followed by the ‘/’symbol and is used to separate the function from x.
Singletons whose membership to a fuzzy set is 0 may be omitted.
2. Union of two fuzzy sets
µAUB(x) = µA(x) V µB(x) = max(µA(x), µB(x))
3. Intersection of two fuzzy sets
µA Intersection B(x) = µA(x) ^ µB(x) = min(µA(x), µB(x))
4. Complement of a fuzzy set
The complement of a fuzzy set A is a new fuzzy set A Complement, containing all the elements which are in the universe of discourse but not in A, with the membership function
Complement of µA(x) = 1 - µA(x)
5. Height of a fuzzy set
The height of a fuzzy set is the highest membership value of the membership function:
Height(A) = max µA(xi)
A fuzzy set with height 1 is called a normal fuzzy set.
In contrast, a fuzzy set whose height is less than 1 is called a subnormal fuzzy set.
6. α-cut of a fuzzy set
α-cut of a fuzzy set A denoted as Aα, is the crisp set comprised of the elements x of a universe of discourse X for which the membership function of A is greater than or equal to α.
Solved problems from various NET papers.
JUNE 2012 – PAPER III Q.No 6
6. If two fuzzy sets A and B are given with membership functions μA(x) = {0.2, 0.4, 0.8, 0.5, 0.1} μB(x) = {0.1, 0.3, 0.6, 0.3, 0.2} Then the value of μ ––– will be A∩B
(A) {0.9, 0.7, 0.4, 0.8, 0.9}
(B) {0.2, 0.4, 0.8, 0.5, 0.2}
(C) {0.1, 0.3, 0.6, 0.3, 0.1}
(D) {0.7, 0.3, 0.4, 0.2, 0.7}
Ans:-A
Explanation:-
The fuzzy intersection of two fuzzy sets A and B on universe of discourse X: μA∩B(x) = min [μA(x), μB(x)] , where x∈X
But here in the question, they are asking for complement of A intersection B and so the answer would be 1-min[A(x),B(x)].
The minimum of 0.2 and 0.1 will be 0.1, and 1-0.1 will be 0.9
The second value is min(0.4,0.3)=0.3 and 1-0.3=0.7
The third value is min(0.8,0.6)=0.6 and 1-0.6=0.4
The fourth value is min(0.5,0.3)=0.3 and 1-0.3=0.7
The last value is min(0.1,0.2)=0.1 and 1-0.1=0.9
The only option which has got the values 0.9,0.7,0.4,0.7 and 0.9, although the fourth value is given as 0.8 instead of 0.7 is option A.
So the answer is option A.
DECEMBER 2012 – PAPER III Q.No 13
13. Consider a fuzzy set A defined on the interval x=[0,10] of integers by the membership function.
µA(x) = x / x+ 2
α cut corresponding to α = 0.5 will be
(A) { 0,1,2,3,4,5,6,7,8,9,10}
(B) {1,2,3,4,5,6,7,8,9,10}
(C) {2,3,4,5,6,7,8,9,10}
(D) { }
Ans:- C
Explanation:-
In the fundamentals, refer to the answer given for question no. 6 regarding α-cut.
α-cut of a fuzzy set A denoted as Aα, is the crisp set comprised of the elements x of a universe of discourse X for which the membership function of A is greater than or equal to α.
Given, x = In the range [0,10]
Membership function = x/x+2
Calculate the value of membership function for the interval from 0 to 10, substituting in the formula x/x+2.
µA(0) = 0 / 0+ 2 = 0
µA(1) = 1 / 1+ 2 = 0.33
µA(2) = 2 / 2+ 2 = 0.5
µA(3) = 3 / 3+ 2 = 0.6
µA(4) = 4 / 4+ 2 = 0.66
µA(5) = 5 / 5+ 2 = 0.71
µA(6) = 6 / 6+ 2 = 0.75
µA(7) = 7 / 7+ 2 = 0.77
µA(8) = 8 / 8+ 2 = 0.8
µA(9) = 9 / 9+ 2 = 0.81
µA(10) = 10 / 10+ 2 = 0.83
α= 0.5. We have to find the corresponding α-cut,
That will be a crisp set, having those values of x, for which the membership function is returning a value of 0.5 or above.
µA(2) = 0.5 and all the values of x above 2 is getting a value greater than 0.5. So the crisp set will contain the following values.
{ 2,3,4,5,6,7,8,9,10}.
So the correct answer is C.
DECEMBER 2013 – PAPER III Q.No 28
28. If A and B are two fuzzy sets with membership functions μA(x) = {0.2, 0.5, 0.6, 0.1, 0.9} μB(x) = {0.1, 0.5, 0.2, 0.7, 0.8} Then the value of μA ∩B
will be
(A) {0.2, 0.5, 0.6, 0.7, 0.9}
(B) {0.2, 0.5, 0.2, 0.1, 0.8}
(C) {0.1, 0.5, 0.6, 0.1, 0.8}
(D) {0.1, 0.5, 0.2, 0.1, 0.8}
Ans:-D
Explanation:-
Intersection of two fuzzy sets
µA ∩B (x) = µA(x) ^ µB(x) = min(µA(x), µB(x))
μA(x) = {0.2, 0.5, 0.6, 0.1, 0.9}
μB(x) = {0.1, 0.5, 0.2, 0.7, 0.8}
μA ∩B={0.1,0.5,0.2,0.1,0.8}
So, the correct answer is D.
29. The height h(A) of a fuzzy set A is defined as h(A) =sup A(x) where x belongs to A. Then the fuzzy set A is called normal when
(A)h(A)=0
(B)h(A)<0
(C)h(A)=1
(D)h(A)<1
Ans:- C
Explanation:-
Explanation:- The height of a fuzzy set is the highest membership value of the membership function:
Height(A) = max µA(xi)
A fuzzy set with height 1 is called a normal fuzzy set.
In contrast, a fuzzy set whose height is less than 1 is called a subnormal fuzzy set. So, according to the above rule, the fuzzy set A is called normal when h(A)=1.
So, the correct answer is 1.
JUNE 2013 – PAPER III Q.No 74
74. If A and B are two fuzzy sets with membership functions μA(x) = {0.6, 0.5, 0.1, 0.7, 0.8} μB(x) = {0.9, 0.2, 0.6, 0.8, 0.5}
Then the value of μ Complement A∪B(x) will be
(A) {0.9, 0.5, 0.6, 0.8, 0.8}
(B) {0.6, 0.2, 0.1, 0.7, 0.5}
(C) {0.1, 0.5, 0.4, 0.2, 0.2}
(D){0.1,0.5,0.4,0.2,0.3}
Ans:- C
Union of two fuzzy sets
µAUB(x) = µA(x) V µB(x) = max(µA(x), µB(x))
μA(x) = {0.6, 0.5, 0.1, 0.7, 0.8}
μB(x) = {0.9, 0.2, 0.6, 0.8, 0.5}
µAUB(x) = {0.9,0.5,0.6,0.8,0.8}
Complement of µAUB(x)={0.1,0.5,0.4,0.2,0.2}
So, the correct answer is C.
JUNE 2014 – PAPER III Q.No 7,8
7. Given U = {1, 2, 3, 4, 5, 6, 7} A = {(3, 0.7), (5, 1), (6, 0.8)} then
~
A will be : (where ~ →complement)
(A) {(4, 0.7), (2, 1), (1, 0.8)}
(B) {(4, 0.3), (5, 0), (6, 0.2) }
(C) {(1, 1), (2, 1), (3, 0.3), (4, 1), (6, 0.2), (7, 1)}
(D) {(3, 0.3), (6.0.2)}
Ans:- C
Explanation:-
Complement of a fuzzy set
The complement of a fuzzy set A is a new fuzzy set A Complement, containing all the elements which are in the universe of discourse but not in A, with the membership function
Complement of µA(x) = 1 - µA(x)
Complement of a fuzzy set A is a new fuzzy set A complement. Since it is a fuzzy set, there will be two members in a singleton. The first member will be all the elements which are in the universe of discourse but not in A. The membership function will be 1- µA(x).
So, the complement of A will be
{(1,1),(2,1),(3,0.3),(4,1),(6,0.2),(7,1)}
The first is (1,1). The first 1 is in U but not in A, so it should be added in the complement. The second 1 is because the membership function is 1- µA(x). 1-0=1.
The same reason why you get (2,1).
The third one (3,0.3) because it is (3,1-0.7)=(3,0.3).
Same reason why you have (4,1) and (7,1).
(6,1-0.8)=(6,0.2).
The member (5,0) is not included because , a singleton whose membership to a fuzzy set is 0, can be excluded .
8. Consider a fuzzy set old as defined below
old={(20,0),(30,0.2),(40,0.4),(50,0.6),(60,0.8),(70,1),(80,1)}. Then the alpha-cut for alpha=0.4 for the set old will be
(A){(40,0.3)}
(B){50,60,70,80}
(C){(20,0.1),(30,0.2)}
(D){(20,0),(30,0),(40,1),(50,1),(60,1),(70,1),(80,1)}
Ans:-D
Explanation:-
alpha-cut of a fuzzy set A will contain those elements where the membership function value is equal to or greater than alpha.
Here, alpha is given a value 0.4. Starting from (40,0.4) all the members have membership function equal or greater than 0.4. so, except
(20,0) and (30,0.2) all the menbers are included in the alpha-cut of the fuzzy set. The only option which has 40,50,60,70, and 80 included is option D. It has
(20,0) and (30,0) too. But it is already noted that any singleton where the membership function is 0 can be considered not included. So basically these two members are not part of the alpha-cut of the fuzzy set A. So the correct option is D.
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OPERATIONS ON FUZZY SETS
