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,

Name: 2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by the
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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