Reasoning Under Uncertainty Questions and Answers

Q: Which situation defeats the Kalman filter's assumptions?

A Gaussian has one mode, so a belief that is genuinely split between two possibilities cannot be represented at all, which is the classic case for particle filtering. High dimension is expensive but tractable, irregular timing is handled by adjusting the transition, and large noise merely widens the covariance. Identifying the failure as representational rather than computational is what makes the choice of filter principled.

Q: Which reasoning pattern does probability handle that logic handles badly?

Probabilistic belief rises and falls with evidence, so a conclusion drawn now can be withdrawn later without any special machinery, whereas monotonic logic needs default rules and truth maintenance to imitate it. Large premise sets, quantification and premise ordering are handled by logic perfectly well. This non-monotonic behaviour, with explaining away as its sharpest example, is the historical reason probability displaced its rivals here.

Q: Let X = [x1, x2, x3], Y = [y1, y2] and z = [z1, z2, z3]. Let R be fuzzy relationy1y2x10.50.1x20.20.9x30.80.6and S be a fuzzy relationz1z2z3y10.60.40.7y20.50.80.9then RoS, by max-min composition yields (rows x1, x2, x3; columns z1, z2, z3) :

The max-min composition gives [0.5 0.4 0.5; 0.5 0.8 0.9; 0.6 0.6 0.7].Each entry is \(\max_{k} \min(R(x_{i}, y_{k}), S(y_{k}, z_{j}))\). For row x1 = (0.5, 0.1):\[ \begin{aligned} z_{1} &: \max(\min(0.5, 0.6), \min(0.1, 0.5)) = 0.5 \\ z_{2} &: \max(\min(0.5, 0.4), \min(0.1, 0.8)) = 0.4 \\ z_{3} &: \max(\min(0.5, 0.7), \min(0.1, 0.9)) = 0.5 \end{aligned} \]The same steps give 0.5, 0.8, 0.9 for x2 = (0.2, 0.9) and 0.6, 0.6, 0.7 for x3 = (0.8, 0.6). Option A already fails at x1z2 (0.5 instead of 0.4); option B takes maxima without the min step, and option D takes minima throughout.

Q: The axiom of excluded middle is represented as :

The law of excluded middle is A ∪ \(\overline{\text{A}}\) = X.For a classical (crisp) set A in the universe X, every element is either in A or in its complement, so the union of A and its complement is the whole universe. Its partner, the law of contradiction, says A ∩ \(\overline{\text{A}}\) = ∅, since nothing is in both.A ∩ \(\overline{\text{A}}\) = X confuses the two laws, and the options built only from \(\overline{\text{A}}\) reduce to \(\overline{\text{A}}\) = X, which holds only for an empty A. In fuzzy set theory both laws fail in general, which is one of the distinguishing points of fuzzy sets.

Q: The centroid method of defuzzification can be represented by the following algebraic expression :

The centroid method is \(z^{*} = \dfrac{\int \mu(z).z\,dz}{\int \mu(z)\,dz}\), option (B), where \(\mu\) stands for \(\mu_{\underset{\sim}{c}}\), the membership function of the aggregated output.The centroid (centre of gravity or centre of area) method takes the crisp output to be the point that balances the area under the membership curve: the integral of \(\mu(z)\) times z, divided by the total area, the integral of \(\mu(z)\).Option (A), the point where membership reaches its maximum, is the max-membership (height) principle. Option (C), a weighted sum over the centres \(\bar{z}\) of the individual output sets, is the weighted average method. Option (D), the midpoint (a + b)/2 of an interval, is the mean of maximum when the maximum is a plateau from a to b.

Q: Functions that qualify as fuzzy intersections and fuzzy unions are referred to as :

They are called t-norms and t-conorms.In fuzzy set theory, a function that can serve as a fuzzy intersection must be a triangular norm (t-norm): commutative, associative, monotonic and with 1 as identity; minimum and the algebraic product are examples. A function that can serve as a fuzzy union must be a triangular conorm (t-conorm, also called s-norm), with 0 as identity; maximum and the algebraic sum are examples.'i-norms' and 'v-conorms' are not standard terms, and pairing t-norms with i-norms leaves out the unions, so the other options are wrong.

Q: A general fuzzy controller consists of .................. .

A general fuzzy controller consists of a fuzzy rule base, a fuzzy inference engine and fuzzification/defuzzification modules.The fuzzification module converts crisp inputs into fuzzy values using membership functions; the inference engine applies the IF-THEN rules of the rule base to them; and the defuzzification module turns the fuzzy output back into a crisp control action.The options naming 'fuzzy functions' or a 'fuzzy data engine' replace a standard component with something that is not part of the architecture.

101.

Which situation defeats the Kalman filter's assumptions?

A.A multimodal belief about the state
B.A state space with a large number of separate dimensions
C.Observations arriving at irregular intervals of time
D.Noise whose variance is larger than expected at the outset

Answer: A

A Gaussian has one mode, so a belief that is genuinely split between two possibilities cannot be represented at all, which is the classic case for particle filtering. High dimension is expensive but tractable, irregular timing is handled by adjusting the transition, and large noise merely widens the covariance. Identifying the failure as representational rather than computational is what makes the choice of filter principled.

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102.

Which reasoning pattern does probability handle that logic handles badly?

A.Conclusions that depend on the order the premises are presented
B.Conclusions that later evidence can retract
C.Conclusions that follow from a very large number of premises
D.Conclusions about individuals drawn from statements about classes

Answer: B

Probabilistic belief rises and falls with evidence, so a conclusion drawn now can be withdrawn later without any special machinery, whereas monotonic logic needs default rules and truth maintenance to imitate it. Large premise sets, quantification and premise ordering are handled by logic perfectly well. This non-monotonic behaviour, with explaining away as its sharpest example, is the historical reason probability displaced its rivals here.

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103.

Let X = [x₁, x₂, x₃], Y = [y₁, y₂] and z = [z₁, z₂, z₃]. Let R be fuzzy relation

	y₁	y₂
x₁	0.5	0.1
x₂	0.2	0.9
x₃	0.8	0.6

and S be a fuzzy relation

	z₁	z₂	z₃
y₁	0.6	0.4	0.7
y₂	0.5	0.8	0.9

then RoS, by max-min composition yields (rows x₁, x₂, x₃; columns z₁, z₂, z₃) :

A.[0.5 0.5 0.5; 0.6 0.8 0.9; 0.5 0.8 0.8]
B.[0.6 0.8 0.9; 0.9 0.9 0.9; 0.9 0.9 0.9]
C.[0.5 0.4 0.5; 0.5 0.8 0.9; 0.6 0.6 0.7]
D.[0.1 0.1 0.1; 0.2 0.2 0.2; 0.5 0.4 0.6]

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: C

The max-min composition gives [0.5 0.4 0.5; 0.5 0.8 0.9; 0.6 0.6 0.7].

Each entry is \(\max_{k} \min(R(x_{i}, y_{k}), S(y_{k}, z_{j}))\). For row x₁ = (0.5, 0.1):

\[ \begin{aligned} z_{1} &: \max(\min(0.5, 0.6), \min(0.1, 0.5)) = 0.5 \\ z_{2} &: \max(\min(0.5, 0.4), \min(0.1, 0.8)) = 0.4 \\ z_{3} &: \max(\min(0.5, 0.7), \min(0.1, 0.9)) = 0.5 \end{aligned} \]

The same steps give 0.5, 0.8, 0.9 for x₂ = (0.2, 0.9) and 0.6, 0.6, 0.7 for x₃ = (0.8, 0.6). Option A already fails at x₁z₂ (0.5 instead of 0.4); option B takes maxima without the min step, and option D takes minima throughout.

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104.

The axiom of excluded middle is represented as :

A.A ∪ \(\overline{\text{A}}\) = X
B.A ∩ \(\overline{\text{A}}\) = X
C.\(\overline{\text{A}}\) ∪ \(\overline{\text{A}}\) = X
D.\(\overline{\text{A}}\) ∩ \(\overline{\text{A}}\) = X

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: A

The law of excluded middle is A ∪ \(\overline{\text{A}}\) = X.

For a classical (crisp) set A in the universe X, every element is either in A or in its complement, so the union of A and its complement is the whole universe. Its partner, the law of contradiction, says A ∩ \(\overline{\text{A}}\) = ∅, since nothing is in both.

A ∩ \(\overline{\text{A}}\) = X confuses the two laws, and the options built only from \(\overline{\text{A}}\) reduce to \(\overline{\text{A}}\) = X, which holds only for an empty A. In fuzzy set theory both laws fail in general, which is one of the distinguishing points of fuzzy sets.

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105.

The centroid method of defuzzification can be represented by the following algebraic expression :

A.\(\mu_{\underset{\sim}{c}}(z^{*}) \ge \mu_{\underset{\sim}{c}}(z)\ \forall z \in Z\)
B.\(z^{*} =\) \(\int \mu_{\underset{\sim}{c}}(z).z\,dz\) / \(\int \mu_{\underset{\sim}{c}}(z)\,dz\)
C.\(z^{*} =\) \(\sum \mu_{\underset{\sim}{c}}(\bar{z}).\bar{z}\) / \(\sum \mu_{\underset{\sim}{c}}(\bar{z})\)
D.\(z^{*} = \dfrac{a + b}{2}\)

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: B

The centroid method is \(z^{*} = \dfrac{\int \mu(z).z\,dz}{\int \mu(z)\,dz}\), option (B), where \(\mu\) stands for \(\mu_{\underset{\sim}{c}}\), the membership function of the aggregated output.

The centroid (centre of gravity or centre of area) method takes the crisp output to be the point that balances the area under the membership curve: the integral of \(\mu(z)\) times z, divided by the total area, the integral of \(\mu(z)\).

Option (A), the point where membership reaches its maximum, is the max-membership (height) principle. Option (C), a weighted sum over the centres \(\bar{z}\) of the individual output sets, is the weighted average method. Option (D), the midpoint (a + b)/2 of an interval, is the mean of maximum when the maximum is a plateau from a to b.

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106.

If \(\underset{\sim}{A}\) = near 0.74 = \(\{ \frac{0}{0.730} + \frac{0.8}{0.735} + \frac{1}{0.740}\) \(+ \frac{0.6}{0.745} + \frac{0}{0.750} \}\)

and \(\underset{\sim}{B}\) = in 0.74 = \(\{ \frac{0}{0.730} + \frac{0.4}{0.735} + \frac{0.8}{0.740}\) \(+ \frac{1}{0.745} + \frac{0.6}{0.750} \}\)

then \(\overline{A \cup B}\) is :

A.\(\{ \frac{1}{0.730} + \frac{0.2}{0.735} + \frac{0}{0.740}\) \(+ \frac{0.4}{0.745} + \frac{1}{0.750} \}\)
B.\(\{ \frac{1}{0.730} + \frac{0.6}{0.735} + \frac{0.2}{0.740}\) \(+ \frac{0}{0.745} + \frac{0.4}{0.750} \}\)
C.\(\{ \frac{2}{0.730} + \frac{0.8}{0.735} + \frac{0.2}{0.740}\) \(+ \frac{0.4}{0.745} + \frac{0.6}{0.750} \}\)
D.\(\{ \frac{1}{0.730} + \frac{0.2}{0.735} + \frac{0}{0.740}\) \(+ \frac{0}{0.745} + \frac{0.4}{0.750} \}\)

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: D

The complement of the union is option D: grades 1, 0.2, 0, 0, 0.4 at 0.730, 0.735, 0.740, 0.745 and 0.750.

For fuzzy sets, the union takes the maximum grade at each point and the complement takes 1 minus the grade.

x	A	B	max	1 - max
0.730	0	0	0	1
0.735	0.8	0.4	0.8	0.2
0.740	1	0.8	1	0
0.745	0.6	1	1	0
0.750	0	0.6	0.6	0.4

Option A has 0.4 and 1 in the last two places, the complement of A alone; option B uses the complement of the intersection in places; and option C has a grade of 2, which is impossible for a fuzzy set.

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107.

\[ \begin{aligned} \underset{\sim}{P} = \Big\{ & \frac{0.1}{C_{1}} + \frac{0.5}{C_{2}} \\ & + \frac{10}{C_{3}} \Big\} \end{aligned} \]

and

\[ \begin{aligned} \underset{\sim}{S} = \Big\{ & \frac{0.3}{S_1} \\ & + \frac{0.8}{S_2} \Big\} . \end{aligned} \]

The relation between \(\underset{\sim}{P}\) and \(\underset{\sim}{S}\) in set theoretic form is :

A.\(\begin{bmatrix} 0.1 & 0.1 \\ 0.3 & 0.5 \\ 0.3 & 0.8 \end{bmatrix}\)
B.\(\begin{bmatrix} 0.1 & 0.3 \\ 0.5 & 0.8 \\ 1 & 0.1 \end{bmatrix}\)
C.\(\begin{bmatrix} 0.2 & 0.5 \\ 0.1 & 0.8 \\ 0.1 & 0.2 \end{bmatrix}\)
D.\(\begin{bmatrix} 0.2 & 0.8 \\ 0.9 & 0.5 \\ 0.3 & 0.3 \end{bmatrix}\)

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

The relation is the first matrix, with rows (0.1, 0.1), (0.3, 0.5) and (0.3, 0.8).

The relation between two fuzzy sets in set-theoretic form is their Cartesian product, whose membership is the minimum of the two memberships: \(\mu_R(C_i, S_j) = \min(\mu_P(C_i), \mu_S(S_j))\). Rows are \(C_1, C_2, C_3\) and columns \(S_1, S_2\).

C1 (0.1): min(0.1, 0.3) = 0.1, min(0.1, 0.8) = 0.1
C2 (0.5): min(0.5, 0.3) = 0.3, min(0.5, 0.8) = 0.5
C3: min with 0.3 and 0.8 gives 0.3 and 0.8

The paper prints the grade of C3 as 10; a membership grade must lie between 0 and 1, so it stands for 1.0, and either way the minimums are 0.3 and 0.8. The other matrices contain values that no minimum of these grades can produce, such as 0.2, 0.9 or 1.

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108.

If the fuzzy relations \(\underset{\sim}{R}\) and \(\underset{\sim}{S}\) are defined as follows :

\[ \underset{\sim}{R} = \begin{bmatrix} 0.7 & 0.5 \\ 0.8 & 0.4 \end{bmatrix} \]

and

\[ \underset{\sim}{S} = \begin{bmatrix} 0.9 & 0.6 & 0.2 \\ 0.1 & 0.7 & 0.5 \end{bmatrix} \]

then using max-min composition T = RoS is represented as :

A.\(\begin{bmatrix} 0.7 & 0.6 & 0.5 \\ 0.8 & 0.6 & 0.4 \end{bmatrix}\)
B.\(\begin{bmatrix} 0.9 & 0.6 & 0.2 \\ 0.1 & 0.7 & 0.5 \end{bmatrix}\)
C.\(\begin{bmatrix} 0.2 & 0.1 & 0.1 \\ 0.7 & 0.1 & 0.1 \end{bmatrix}\)
D.\(\begin{bmatrix} 0.2 & 0.3 & 0.4 \\ 0.1 & 0.1 & 0.9 \end{bmatrix}\)

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

The composition is the first matrix, with rows (0.7, 0.6, 0.5) and (0.8, 0.6, 0.4).

In max-min composition, \(T(x, z) = \max_y \min(R(x, y), S(y, z))\): take the minimum of each pair along a row of R and a column of S, then the maximum of those minimums.

Row 1: max(min(0.7, 0.9), min(0.5, 0.1)) = 0.7; max(0.6, 0.5) = 0.6; max(0.2, 0.5) = 0.5
Row 2: max(min(0.8, 0.9), min(0.4, 0.1)) = 0.8; max(0.6, 0.4) = 0.6; max(0.2, 0.4) = 0.4

The second option simply repeats S, and the last two begin with 0.2 where the composition gives 0.7, so they do not follow the max-min rule.

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109.

Functions that qualify as fuzzy intersections and fuzzy unions are referred to as :

A.t-norms & t-conorms
B.i-norms & v-conorms
C.t-norms & i-norms
D.None of the above

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

They are called t-norms and t-conorms.

In fuzzy set theory, a function that can serve as a fuzzy intersection must be a triangular norm (t-norm): commutative, associative, monotonic and with 1 as identity; minimum and the algebraic product are examples. A function that can serve as a fuzzy union must be a triangular conorm (t-conorm, also called s-norm), with 0 as identity; maximum and the algebraic sum are examples.

'i-norms' and 'v-conorms' are not standard terms, and pairing t-norms with i-norms leaves out the unions, so the other options are wrong.

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110.

A general fuzzy controller consists of .................. .

A.Fuzzy rule base, fuzzy inference engine and fuzzification/defuzzification modules
B.Fuzzy functions, fuzzy inference engine and fuzzification/defuzzification modules
C.Fuzzy rule base, fuzzy data engine and fuzzification/defuzzification modules
D.Fuzzy functions, fuzzy data engine and fuzzification/defuzzification modules

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: A

A general fuzzy controller consists of a fuzzy rule base, a fuzzy inference engine and fuzzification/defuzzification modules.

The fuzzification module converts crisp inputs into fuzzy values using membership functions; the inference engine applies the IF-THEN rules of the rule base to them; and the defuzzification module turns the fuzzy output back into a crisp control action.

The options naming 'fuzzy functions' or a 'fuzzy data engine' replace a standard component with something that is not part of the architecture.

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