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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

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.

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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

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.

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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

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.

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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.

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.

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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

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.

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116.

Let A be the set of comfortable houses given as

A = { \(\frac{x_{1}}{0.8}\), \(\frac{x_{2}}{0.9}\), \(\frac{x_{3}}{0.1}\), \(\frac{x_{4}}{0.7}\) }

and B be the set of affordable houses

B = { \(\frac{x_{1}}{0.9}\), \(\frac{x_{2}}{0.8}\), \(\frac{x_{3}}{0.6}\), \(\frac{x_{4}}{0.2}\) }

Then the set of comfortable and affordable houses is

Exam Year(s)

UGC NET - 2013,

Answer: A

The set of comfortable and affordable houses is the fuzzy intersection of A and B, which takes the minimum of the two membership values for each house.

\[ x_{1}: \min(0.8, 0.9) = 0.8 \]

\[ x_{2}: \min(0.9, 0.8) = 0.8 \]

\[ x_{3}: \min(0.1, 0.6) = 0.1 \]

\[ x_{4}: \min(0.7, 0.2) = 0.2 \]

So the set is x1/0.8, x2/0.8, x3/0.1, x4/0.2, the first option. The second and third options use maxima for some houses, as in a union, and the fourth is not a min or max of the given values.

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117.

Support of a fuzzy set

A = { \(\frac{x_{1}}{0.2}\), \(\frac{x_{2}}{0.15}\), \(\frac{x_{3}}{0.9}\), \(\frac{x_{4}}{0.95}\), \(\frac{x_{5}}{0.15}\) }

within a universal set X is given as

Exam Year(s)

UGC NET - 2013,

Answer: D

The support of a fuzzy set is the crisp set of all elements whose membership value is greater than zero. Every element x1 to x5 has a membership above zero (0.2, 0.15, 0.9, 0.95 and 0.15), so the support is \(\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\}\).

The set \(\{x_{3}, x_{4}\}\) would be the elements with a high membership, such as an alpha-cut at 0.9, and not the support. The first two options show fuzzy sets with equal membership values, which are not the definition of the support.

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118.

Given \(U = \{1, 2, 3, 4, 5, 6, 7\}\), \(A = \{(3, 0.7), (5, 1), (6, 0.8)\}\) then \(\tilde{A}\) will be : (where \(\sim \rightarrow\) complement)

Exam Year(s)

UGC NET - 2014,

Answer: C

\(\tilde{A}\) is {(1, 1), (2, 1), (3, 0.3), (4, 1), (6, 0.2), (7, 1)}.

The complement of a fuzzy set has membership \(1 - \mu_A(x)\) for every element of the universe. So 3 gets \(1 - 0.7 = 0.3\), 6 gets \(1 - 0.8 = 0.2\), 5 gets \(1 - 1 = 0\) and so is not listed, and the elements 1, 2, 4 and 7 that are not in A have membership 1.

Options A and B list wrong elements or memberships, and option D keeps only elements that already belong to A and omits all elements outside A.

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119.

Consider a fuzzy set old as defined below Old = {(20, 0.1), (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

Exam Year(s)

UGC NET - 2014,

Answer: D

The alpha-cut for alpha = 0.4 is given by option D, the crisp set that has 1 for every element with membership at least 0.4.

The elements 40, 50, 60, 70 and 80 have memberships 0.4, 0.6, 0.8, 1 and 1, all at least 0.4, so they get membership 1, and 20 and 30 have 0.1 and 0.2, so they get 0. This is the set written in option D.

Option A keeps only one element, option B lists the elements with membership above 0.4 only, without memberships, and option C gives the elements below 0.4, which is the opposite.

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