2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by themembership functions1pHa(x) =+10"1H(x) =10*Hi(x)x+1Determine mathematical membership functions and graphs of each ofthe followings :а) AU B, BоC,b) AUBUC, AOBOCc) Anī, BUCd) AOB, ĀU B2.2 Show the two fuzzy sets satisfy the De Morgan's Law.1HA(x) =1+(x-10)1HB(x) =1+x2.3 Show that the following sets don't satisfy the law of contradiction andthe law of excluded middle.a) H,(x) =-1+x60 / 340b) 4 3D {(а, 0.4), (b, 0.5), (с, 0.9), (d, I)}2.4 Determine complements, unions, and intersections of the followingsets by using Yager's operators for o = 1, 2a) A = {(a, 0.5), (b, 0.9), (c, 0.1), (d, 0.5)}1b) дл(х) —.1+x2.5 Compute the complements of the following sets by Yager'scomplements w = 1, 2.а) А — {(а, 0.5), (b, 0.9), (с, 0.3)}1b) H,(x) =1+(x–1)522. The Operation of Fuzzy Set2.6 Compute the complements of the following sets by using Probabilistic,Bounded, Drastic, and Hamacher product.1a) Ha(x) =-1+x?b) Ha(x) = 2*с) А - {(а, 0.4), (b, 0.5), (с, 0.9)}2.7 Compute the simple disjunctive sum, disjoint sum, simple difference,and bounded difference of the setsА 3 {(х, 0.5), (у, 0.4), (z, 0.9), (w, 0.I)}В 3 {(х, 0.4), (у, 0.8), (z, 0.1), (w, 1)}2.8 Determine the distances (Hamming, Euclidean and Minkowski for @ =2) between the following setsA = {(x, 0.5), (y, 0.4), (z, 0.9), (w, 0.1)}B = {(x, 0.1), (y, 0.9), (z, 0.1), (w, 0.9)}2.9 Prove the following properties.a) Let function Ta t-norm operation.The following T'is a t-conorm operation.т (х, у) — 1-т(1-х, 1-у)b) х1у%3D хту аnd xту-xlуwherex =1-x, j=1-y and xTy=1+ (x, y)where I represents t-conorm operator.2.10 Determine the closet pair of sets among the following setsА 3 {(х, 0.4), (х, 0.4), (х, 0.9), (х, 0.5)}В 3 {(х, 0. 1), (х, 0.0), (х, 0.9)}С - {(х, 0.5), (х, 0.5), (х, 0.9)}2.11 Show the Max operator satisfies the properties boundary condition,

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Description: 2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by themembership functions1pHa(x) =+10"1H(x) =10*Hi(x)x+1Determine mathematical membership functions and graphs of each ofthe followings :а) AU B, BоC,b) AUBUC, AOBOCc) Anī, BUCd) AOB, Ā

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2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by the membership functions 1 · H(x) 1 Ha(x) Ha(x) x+1 x² +10 10" Determine mathematical membership functions and graphs of each of the followings : a) AU B, ВоС, b) AUBUC, AOBOC c) Anī, BUC d) AnB, ĀU Ē

2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by the membership functions 1 · H(x) 1 Ha(x) Ha(x) x+1 x² +10 10" Determine mathematical membership functions and graphs of each of the followings : a) AU B, ВоС, b) AUBUC, AOBOC c) Anī, BUC d) AnB, ĀU Ē

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sequence and Series

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2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by the
membership functions
1
pHa(x) =+10"
1
H(x) =
10*
Hi(x)
x+1
Determine mathematical membership functions and graphs of each of
the followings :
а) AU B, BоC,
b) AUBUC, AOBOC
c) Anī, BUC
d) AOB, ĀU B
2.2 Show the two fuzzy sets satisfy the De Morgan's Law.
1
HA(x) =
1+(x-10)
1
HB(x) =
1+x
2.3 Show that the following sets don't satisfy the law of contradiction and
the law of excluded middle.
a) H,(x) =-
1+x
60 / 340
b) 4 3D {(а, 0.4), (b, 0.5), (с, 0.9), (d, I)}
2.4 Determine complements, unions, and intersections of the following
sets by using Yager's operators for o = 1, 2
a) A = {(a, 0.5), (b, 0.9), (c, 0.1), (d, 0.5)}
1
b) дл(х) —.
1+x
2.5 Compute the complements of the following sets by Yager's
complements w = 1, 2.
а) А — {(а, 0.5), (b, 0.9), (с, 0.3)}
1
b) H,(x) =
1+(x–1)
52
2. The Operation of Fuzzy Set
2.6 Compute the complements of the following sets by using Probabilistic,
Bounded, Drastic, and Hamacher product.
1
a) Ha(x) =-
1+x?
b) Ha(x) = 2*
с) А - {(а, 0.4), (b, 0.5), (с, 0.9)}
2.7 Compute the simple disjunctive sum, disjoint sum, simple difference,
and bounded difference of the sets
А 3 {(х, 0.5), (у, 0.4), (z, 0.9), (w, 0.I)}
В 3 {(х, 0.4), (у, 0.8), (z, 0.1), (w, 1)}
2.8 Determine the distances (Hamming, Euclidean and Minkowski for @ =
2) between the following sets
A = {(x, 0.5), (y, 0.4), (z, 0.9), (w, 0.1)}
B = {(x, 0.1), (y, 0.9), (z, 0.1), (w, 0.9)}
2.9 Prove the following properties.
a) Let function Ta t-norm operation.
The following T'is a t-conorm operation.
т (х, у) — 1-т(1-х, 1-у)
b) х1у%3D хту аnd xту-xlу
where
x =1-x, j=1-y and xTy=1+ (x, y)
where I represents t-conorm operator.
2.10 Determine the closet pair of sets among the following sets
А 3 {(х, 0.4), (х, 0.4), (х, 0.9), (х, 0.5)}
В 3 {(х, 0. 1), (х, 0.0), (х, 0.9)}
С - {(х, 0.5), (х, 0.5), (х, 0.9)}
2.11 Show the Max operator satisfies the properties boundary condition,

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