7. Suppose A is denumerable. Prove that there is a partition P of A such that P is denumerable and for every X E P, X is denumerable.

Q: Let P be a partition on [a,b]. Then ƒ € R[a, b] if and only if inf > sup f.Ax k=1 [xk-1.Xk] sup 2,…

A: To check if the given statement is true or false

Q: Prove (sn) is bounded if and only if lim sup |Sn| < +∞.

A:

Q: (a) Let A = { 1.r €R. Show that A is a bounded subset of R. %3D 2x2+1 Hence find inf A and sup A.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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Q: Suppose f [a, b] → R is integrable with the property that L(f, P₁) = L(f, P₂) for all pairs of…

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Q: Consider the set A = {1/2^{n-1}: n\epsilon \mathbb{N}}. Show that Inf(A) = 0. WITHOUT USING LIMITS

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Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

Q: Prove that if KCR is compact and non-empty, then sup(K) exists and sup(K) E F

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Q: Let n be a positive integer. The coefficient of x°y° in the expansion of (1+ xy+ y²)' equals

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Q: in in Signed-Rank Test a two-sided alternative, we may reject H0 in favor of H1 if either w+ or w-,…

A: Signed rank test for  two tailedHo:M≠Mowe reject Ho, if W+≤Wα and W-≤Wα

Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: Use Simpsons 1/3 rule to evaluate of e-x2 over the interval 0.2 to 1.5 using 6 segments ?

A:

Q: Let M and N be nonempty bounded subsets of R and let M + N := {m + n : m ∈ M and n ∈ N} . Prove that…

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Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

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Q: For allne z, gcdin,nt)=l.

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Q: 1. Prove that if A is infinite and a e A, then A – {a} is also - infinite.

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Q: In the domain of Gaussian integers Zli), the element 19 is * O Reducible O Unit Irreducible

A: Answer is irreducible. Proof is given below.

Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: 2. Recall that the characteristic function of a set E is defined as I is in E 1, Xe(z) = { 0, z is…

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Q: If ø + SCR is bounded above, then .S has a supremum إختر واحداً: ibs

A: I have proved by contradiction

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: (2) Show that if dim(V) = k (so in particular, T has k distinct cigenvalues), then there exists v e…

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Q: - Prove thas the scries 2+x2 3+X2 4+x niformcy conuergen2 in any interval?

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Q: For for any pestive integor n Show that %3DFHE nth fibonacci term fn=the

A: We need to show that , for any positive integer n ,  f1 f2 + f2 f3 + ....... + f2n-1 f2n = f2n2…

Q: 12.- Suppose K and F are disjoint in RP where K is compact and F is closed. Prove that there exists…

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Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

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Q: (X,) is a sequence of independent identically distributed positive RVs, where E(X,) = a and EX,'=b…

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Q: 1. Prove that fraction where m is z-4

A: Given: zn-1z-1 ⇒zn-1nz-1 ----(1) Let apply the limit in equation (1) limz→1zn-1nz-1 ----(2) Let n=pq…

Q: 8. (a) Suppose f : [a, b] → R is integrable and L(ƒ, P) = U(ƒ, P) for some partition P of [a, b].…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: Compute the convolution x[n]*h[n] for all n> 0, by using Graphical or Tabular method. x[0] = 3, x[1]…

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Q: Show that cl(a) = {a} if and only if a ∈ Z(G).

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Q: Prove or disprove: If xn ≥ 0 for all n ∈ N and lim sup xn = 0, then xn → 0

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Q: Let P = {X E P(Z*)|X is finite}. Prove that P is denumerable.

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Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

Q: 9.K Let X = Q y K = {p € Q : 2 < p? < 3} . Show that K is closed in Q, is bounded, and is not…

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Q: 2. Suppose that {v1,... , Vp} is an orthonormal set in R", and that x E R". What can you say about…

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Q: 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = 0.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 6. Let K be compact in R and let c E R. Prove that there exists a point a € K such that |c – a| =…

A: Given- Let K be compact in R and let c ∈ R.  To Prove- Prove that there exists a point a ∈ K such…

Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: Let o,T E Sn for some positive integer n satisfy o(k) 1< k < n. Prove that o-(k) = k and p(k) = k…

A: Solution: Let us consider the symmetric group Sn, for some positive intiger n.  We know that Sn is…

Q: (a) Suppose ƒ: [a, b] → R is integrable and L(f, P) = U(ƒ, P) for some partition P of [a, b]. What…

A: (a) Given that f:a,b→ℝ is integrable and Lf,P=Uf,P for some partition P of [a, b]. To find: what f…

Q: Let x, y, z ∈  ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z

A: In the given question we have to prove that if x|yz , then x|z and g.c.d(x,y)=1 and x,y,z∈ℕ. we will…

Q: In a discrete space X O every subset of X is dense in X the only dense subset of X is X O a proper…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: In Z[V-5], the element 29 is: None of the choices Irreducible Unit reducible

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Q: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

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Q: 4. Find the number of surjections from {1, 2, ..., m} to {1, 2, .. ., n} such that no element in the…

A: We have to find the number of Surjection from one set to another set.

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below