Lemma: Let P and O be two partitions of !=[a,b] such that Ps0 Then, prove that US.QSUU.P).

Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

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Q: Let U=(1, 2, 3, 4, 5, 6, 7), A=(1, 3, 5, 7), and B=(2, 5, 7). Find the set AnB AnB= (Use ascending…

A: Note: The complement of set A is defined as a set that contains the elements present in the…

Q: A nB oiaivalar 111222719;10s b) A'U (B n C) 9. Find the number of the subsets and list all subsets…

A: The number of subsets of a set with n elements is 2n.

Q: 4: Ex 4.) Let S be a non empty bounded set in R. a ( a. Let azo, and let oS := {as: SES}. Prove that…

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Q: Let R be the set of all real numbers. Is R*, R , {0}} a partition of R? Why or why not? Select- v.…

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Q: Let U= {1, 7, 6, 14, 5, 10, 20, 18) A = {1, 20, 18} C = {7, 6, 14, 5, 10} Find A U C'.

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Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: Show this collection of subsets a partition of A UBUC {A – (B U C), B – (A U C), C – (A U B), ANBN…

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Q: Find the power set of: a) {w, {x}, {y, z}} b) {1,2,7} с) (3,(1,4)}

A: Given:-  {w, {x}, {y, z}}  {1,2,7}  {3,{1,4}} To find:- power set  of given sets

Q: if JERLA.b] and it hex)- J(x). except for finite mumber of Paint in Cab] then that he Rlab] prove.…

A: if g ∈R[a,b] and if h(x) = g(x) except for a finite number of points in [a,b] then prove that h…

Q: Assume that E, F, G₁, G₂, G., G (wherein i = 1 ton) such that P(EG) is less than or equal to P(FIG)…

A: To show that P(E) ≤ P(F), we can use the law of total probability. We know that the G's form a…

Q: Both questions below refer to set A = Z – {-2,0}. a) Give a partition of A into infinite many…

A: Solution: Definition: Let Ψ be a partion of the set A. That means, Ψ is the collection of subsets of…

Q: Find an eguaton for the set of all points (a sy, z) equiditant fom point (1,0,2) and

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Q: 3) Jet Oxln) = #* pernietations o [n] with order k. a) Show thot e

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Q: şa,6şc2.Debine a selation I on. pl2) which satisfies. For all set A and B in P(X) AJB ANB+

A: Given x={a, b, c}P(x)={ {a} , {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}, ∅}a relation onJ: P(x)…

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: O lalvite eah vertar vertors in S. (if not posi ble, enter IMpossiecE) a inecr Com bimtion of the as…

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Q: Least upper bound (lub) ond gueatest lowr bound 1.b) af the set are.

A: Just explore the set and look for biggest and smallest elements of given set.

Q: 6x:-ld 212CK) 's aving of integer num 12 and fo), gaJE Z, ) such That 9Cx) = Prove Fhat dey (fagx))…

A: Given  Ring Z12(x)  f(x) = 2+5x+4x4  g(x) = 1+x+x2+3x3    So clearly,  degree of f(x) = 4  degree of…

Q: Consider (Power set of R, c). Let B= {{3},N,Z,Q,R-Q), then Sup(B)= %3D O a. R O b. Q O c. R-Q Od.…

A: Consider Power set of R, ⊆ Let B=3, N, Z, Q, R-Q then SupB is the set S∈Power set of R such that…

Q: Buniakovski26 inequality) If neN and a,a2, ... ,an, b,b2,..,bn are real numbers, prove that k%3D1…

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Q: OShow that the set (1,2] is not compact 2Show that the set [1,2) is not compact 3Show that the set…

A: 1.To show that (1,2] is not compact

Q: let it be T=11,2,3 } S:la,b,c] Proved P(TNS)E P(S)NP(T)

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Q: Suppose ACB and BCC. Show that = {A, B-A, C-B} is a partition of C.

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Q: Let P* be a refinement of a partition, P. Show U(P*, f, a) < U(P, f, a)

A: Upper Riemann sum and lower Riemann sum are the very important parts of Riemann integration. Lower…

Q: If a, <a, then prove that A2A, where denotes a crisp superset (i.e., suppositions) relation.

A: The concept of alpha-cuts (α-cuts) in fuzzy set theory represents the crisp set of all the elements…

Q: Let FCA) be the nwnber of and let art) be the nmber men Of women residingina certain town year t Let…

A: *given :- total number of men in year t, f(t) = 250 + 9t total number of women in year t, g(t) = 275…

Q: find the subset of (E, B,T,*)

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Q: Let A=(aij)nxn be symmetric and positive definite. Show that laul< (anan), , lis = 1,2,..n 9

A: Given: Let us consider the given, A=aijn×n Let's prove that aij&lt;aiiajj12,i,j=1,2,...,n.