Reasoning Under Uncertainty Questions and Answers

111.

Consider a fuzzy set A defined on the interval X = [0, 10] of integers by the membership Junction

\(\mu_A(x) = \frac{x}{x+2}\)

Then the \(\alpha\) cut corresponding to \(\alpha\) = 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.{ }

Exam Year(s)

UGC NET - 2012,

Answer: C

The alpha cut of a fuzzy set contains the elements whose membership is at least alpha. Here the membership is \(\mu_A(x) = x/(x+2)\) and alpha = 0.5, so the condition is

\[ \frac{x}{x+2} \ge 0.5 \]\[ x \ge 0.5x + 1 \Rightarrow x \ge 2 \]

The values of x in the interval [0, 10] that satisfy this are 2, 3, ..., 10. For x = 1 the membership is 1/3, which is less than 0.5, and for x = 0 it is 0, so these are left out. The set is {2, 3, 4, 5, 6, 7, 8, 9, 10}, the third option. The empty set would hold only if no value reached 0.5.

Discuss in Forum

112.

If A and B are two fuzzy sets with membership functions

\(\mu_{A}(x) = \{0.2, 0.5, 0.6, 0.1, 0.9\}\)
\(\mu_{B}(x) = \{0.1, 0.5, 0.2, 0.7, 0.8\}\)

Then the value of \(\mu_{A \cap B}\) 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}

Exam Year(s)

UGC NET - 2013,

Answer: D

The intersection of two fuzzy sets takes the minimum of the membership values element by element, \(\mu_{A \cap B}(x) = \min(\mu_{A}(x), \mu_{B}(x))\). The pairs are (0.2, 0.1), (0.5, 0.5), (0.6, 0.2), (0.1, 0.7) and (0.9, 0.8), whose minima are 0.1, 0.5, 0.2, 0.1 and 0.8.

So the result is {0.1, 0.5, 0.2, 0.1, 0.8}. Option A takes maxima, which is the union, and options B and C each take a wrong value at one or more positions.

Discuss in Forum

113.

The height h(A) of a fuzzy set A is defined as

\(h(A) = \sup_{x \in A} A(x)\)

Then the fuzzy set A is called normal when

A.\(h(A) = 0\)
B.\(h(A) \lt 0\)
C.\(h(A) = 1\)
D.\(h(A) \lt 1\)

Exam Year(s)

UGC NET - 2013,

Answer: C

A fuzzy set is called normal when its height h(A) equals 1, that is, at least one element has full membership. The height is the supremum of the membership values, so a set with h(A) = 1 reaches the maximum possible grade.

If h(A) is 0, or less than 0 or less than 1, no element has full membership, so the set is sub-normal. A value below 0 is impossible for a membership function, which lies in the interval from 0 to 1.

Discuss in Forum

114.

A fuzzy set A on R is ________ iff \(A(\lambda x_{1} + (1 - \lambda)x_{2}) \ge \min [A(x_{1}), A(x_{2})]\) for all \(x_{1}, x_{2} \in R\) and all \(\lambda \in [0, 1]\), where min denotes the minimum operator.

A.Support
B.\(\alpha\)-cut
C.Convex
D.Concave

Exam Year(s)

UGC NET - 2013,

Answer: C

A fuzzy set A on R is convex if and only if \(A(\lambda x_{1} + (1 - \lambda)x_{2}) \ge \min [A(x_{1}), A(x_{2})]\) for all \(x_{1}, x_{2}\) and all \(\lambda \in [0, 1]\), which is option (C). This says that the membership of any point between two points is at least the smaller of their memberships.

The support is the set of points with non-zero membership, an \(\alpha\)-cut is the set with membership at least \(\alpha\), and concave is not the term for this property. So (A), (B) and (D) are wrong.

Discuss in Forum

115.

If A and B are two fuzzy sets with membership functions
\(\mu_{A}(x) = \{0.6, 0.5, 0.1, 0.7, 0.8\}\)
\(\mu_{B}(x) = \{0.9, 0.2, 0.6, 0.8, 0.5\}\)
Then the value of \(\mu_{\overline{A \cup B}}(x)\) 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}

Exam Year(s)

UGC NET - 2013,

Answer: C

The membership of the complement of \(A \cup B\) is found by taking the maximum of the two memberships point by point, and then subtracting from 1:

\[ \mu_{A \cup B} = \{0.9, 0.5, 0.6, 0.8, 0.8\} \]

\[ 1 - \mu_{A \cup B} = \{0.1, 0.5, 0.4, 0.2, 0.2\} \]

This is option (C). Option (A) is the union itself, option (B) is the point by point minimum, which is the intersection, and option (D) differs in the last value, which would need a membership of 0.7.

Discuss in Forum

Loading…

Topics

Other Subjects

Tags: Reasoning Under Uncertainty Questions and Answers || Reasoning Under Uncertainty MCQ Questions and Answers || Reasoning Under Uncertainty GK Questions and Answers || Reasoning Under Uncertainty GK MCQ Questions || Reasoning Under Uncertainty Multiple Choice Questions and Answers || Reasoning Under Uncertainty GK || GK on Reasoning Under Uncertainty || Artificial Intelligence Questions and Answers || Artificial Intelligence MCQ Questions and Answers || Artificial Intelligence GK Questions and Answers || GK on Artificial Intelligence

Reasoning Under Uncertainty Questions and Answers

Topics

Other Subjects

About Us

Stay informed about all important Dates & Notifications