**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.**