Suppose that E is an alphabet, and that f :E* → E* has the property that f(ơ) = o for every o e E an f(xy) = f(x) f(y) for every x, y € £*. Prove that for every x € E*, f (x) = x.

Q: Show that the Stein operator for Poisson P(Y = k) = is Tf(r) = Af(k + 1) – kf(k).

A: Given information: The Stein operator for Poisson P(Y = k) = λk/k! is Tµf(x) = λf(k + 1) – kf(k).

Q: Let D be a subset of R. Show that f : D → R is uniformly continuous on D if and only if ƒ satisfies…

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Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: If the Wronskian W of f and g is t'et, and if f(t) = t, find g(t). NOTE: Use c as an arbitrary…

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Q: 1. Suppose that f(2) is analytic on the set A = {z :1< |z| < 2}, and let f(2) = E apzk be its…

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Q: Prove that S4 is not isomorphic to D12.

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Q: Let D be a subset of R. Show that f : D → R is uniformly continuous on D if and only if ƒ satisfies…

A: According to the given information, it is required to explain that:

Q: Define a mapping fn : Z → Z by fn(x) = 1 – nx for each integer n E Z. a. For which values of n is fn…

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Q: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: Let X e L'(N, F,P) and let G,H be sub o-algebras of F. Moreover be independent of o(o(X),G). Show…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: Let f: (X.d) - (Y,di) be continuous fumetion, then for any open set Gin X. f(G) is not neressar y…

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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: If the Wronskian W of f and g is 3e^(4t), and f(t) = e^(2t), find g(t)

A: Wronskian W of f and g is given by:

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find Z_2×Z_2.

Q: Let S be a non empty subset of R and let u e R be such that u + is an upper bound of S for each n e…

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Q: Suppose that E is an extension of F of prime degree. Show that, forevery a in E, F(a) = F or F(a) =…

A: Let E be an extension of F of degree p (prime) =&gt; [E : F] = p Here we will use tower property to…

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Show that the Modulus Function            f : R → R, given by f(x) = | x |, is neither one-one nor…

A: Consider, f: R → R given by f(x) = | x |Where,| x | is x, if x is positive or 0| x | is – x, if x is…

Q: The Wronskian of y,(t)=e 2, y,()=e" is-e St. Select one: True False

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Q: 1. For every element x in A, there is an element y in B such that (x, y)EF

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Suppose I is an open interval and f € C°(I). Show by induction that ffor х, а € 1, f(x) = f(a) +…

A: Given: I is an open interval and f∈C∞I   To show: fx=fa+∑k=1nfkak!x-ak+1n!∫axx-tnfn+1tdt for x,a∈I…

Q: Let h : N x N → N be the function h(x, y) = 3x + 2y. Use strong induction to prove that if n EN and…

A: Given that h : ℕ×ℕ→ℕ be the function h(x,y)=3x+2y. Then we have to prove by using strong induction…

Q: 1. ( Prove that for any x, y e R, we have ||| - ||y|| < |× -y|l. 2. Let u e R. Prove that 0 u 0,…

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Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: If f, g E R[a,b] with f < g on [a,b], prove that g.

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Q: Let K be an extension of be an algebric element o 2 F Show that a ² is algebr

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Q: (b) Define the operator T: → as: (Tx) = for every x = () e t. That is, Tx = (61. for every x (f,E2.…

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Q: To be onto, for any integer y, there exists an integer x such that y = f(x). Give a counterexample,…

A: To be onto for any real number y, there exists a real number x such that y = f(x). Let y be a real…

Q: Let H(x):=k for x=, (keN) and H(x):=0 elsewhere on [0,1] Show that H(x) 4ER[0,1].

A: In this question, we use the definition of Riemann integral to check the function H(x) is Riemann…

Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

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Q: Find the smallest integer in the given set a.{ xϵ Z f x &gt; 0 and x = 4s + 6t for some s, t in Z }

A: Consider the equation: x=4s+6t……1   where: x, s, t∈ℤ and x&gt;0

Q: Let A be a variation of the Ackermann function defined by 2n if m = 0 if m >1 and n = A(m, n) || 2…

A: Given that A is a variation of the Ackermann function defined by,A=2nif m=00if m≥1 and n=02if m≥1…

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

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Q: Suppose  there exist transposition f_i , 1<=I<=t  such that

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Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: In the proof that |(0, 1)| > |N|, we use Cantor's Diagonal Method, where we change the nth digit dnn…

A: We are given the following. In the proof that |(0, 1)| &gt; |N|, we use Cantor's Diagonal Method,…

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: If m, n are relatively prime, then o(mn) = o(m)o(n)

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Q: (a) Let X = {m,n, p, q}, and = {ø, X, {m},{n}, {p, q}, {m,n},{n,p, q}, {m, p, q} %3D a sigma algebra…

A: Given: Let X=m, n, p, q and Σ=∅, X, m, n, p, q, m, n, n, p, q, m, p, q. (a) Find whether Σ a sigma…

Q: (12) If E(X*) is finite with E(X) = µ and Var(X) =o² then prove that E((X – µ)*) > of. %3D

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

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: Let SCR and let c e R be a cluster point of S. Prove that for each e >0 the set W := Sn(c-e,c+e) has…

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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: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

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Q: Let A ∈ Mn×n(F). Under what conditions is det(−A) = det(A)?

A: In determinant the scalar quantity is taken common from a single row or a column.

Q: Define a set X of numbers as follows. В. 2 E X. R2. If x E X, so is x+ 10. Use induction to prove…

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Q: 1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a…

A: (a) Given that  f:X→Yg:Y→Z and g∘f is onto To prove : g is onto Let x∈X, y∈Y , z∈Z As gof is onto…

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

A: The complete solutions are given below