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: Define f(n) = +¹-(-¹)" for all n € Z. Thus, ƒ : Z → ZZ. Which is correct? 4 Select one: a. f is not…

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Q: This is a real analysis question. For each n ∈ N, suppose that the functions fn : [−2,2] → [0,1]…

A: For each n ∈ N, suppose that the functions fn : [−2,2] → [0,1] are continuous and have the property…

Q: Show that the function a which is defined on S3 with x α = x =1 is not homomorphism.

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Q: Let f and g be the following functions. f: N→ P(N) - {0} defined by f(x) = {n: n € N and x ≤ n ≤…

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Q: If f is [Select] f(x) > [Select] [Select] on (a, b) and f(x) #0 for all x in (a, b), then either for…

A: This is a question on the properties of continuous function.

Q: 3. Suppose that f:A→B and g:B→C are uniformly continuous. Prove that the composition g(f):A-C is…

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Q: Let A be a set and let a € A. Suppose that f: A→ A satisfies (a) a = f (a), (b) if x Ey then f (x) =…

A: Given: f:A→A a) a∈fa b) fx∈fy To find: Solution

Q: Show that for all x, e* > 1 +x.

A: It is given that, ex≥1+xwe have to show that for all x.

Q: Prove the given theorem:  If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof) =…

A: Given that

Q: For any k > 0 E Z, show that there is a function f : Z* → Z which is both one-to-one and onto. (Here…

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Q: Prove that any entire function is identically constant on C if if either one of the following…

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Q: Suppose that f: Z→ Z is one-to-one. Let's define g: Z→ Z² by g(n) = (\n\, f(n)|n|). Prove that g is…

A: The answer provided below has been developed in a clear step by step manner.

Q: 8. A function g : Z → Z is defined by g(n) = 3n – 2. Let D be the set of odd integers. (a) Determine…

A: The objective is to solve the above question.

Q: If f(x) ∈∈ c[a, b]  and f(a)* f(b) <0 , then there exist   c∈∈ (a,b) such that :

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Q: Suppose the function f : R → R defined by f(x) = [x]. Select all correct statements (there could be…

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Q: et H={Φ∈Sn | Φ(n)=n}. Find index of H in Sn.

A: H=ϕ|ϕ(n)=n=I (group of identity permutation)o(H)=1o(Sn)=n!index of H in Sn=n!1=n!

Q: Consider the function f : N → N defined by n? if n is odd f = 2n + 1 if n is even A. Find f(A) if A…

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Q: Suppose f : A →→ B is an injective function. Show that there is a function g : B→ A such that gof=…

A: Let's prove this statement step by step. Want to show that for any injective function f:A→B, there…

Q: 2. If X is a finite set and fn → f on X, show that fn = f on X. 3. If fn 3 f on X and Y is a…

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Q: 6. Use the definition to show: if fn →f uniformly on S and 9n → g uniformly on S, then fn +9n → f +…

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Q: Suppose f: R→→ R is convex. Show that for all x € [a, b]. x-a :) ≤ b = = f(a) + ² = f(b) b a f(x) ≤

A: Suppose   is convex. We have to show that:

Q: Consider a function f: A→ B. Suppose there exists a function g: B→ A such that the composition go f…

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Q: Let f:R → R given by f(x) = 2x² and A = [-3, 1] U [2, 4]. Then f(A) is

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Q: The functions f : N → N and g : N² → N are recursively defined as follows, where n and m are each…

A: g(m, n) = g(m-1, n) + 2n             = g(m-2, n) + 2n + 2n            = g(m-3, n) + 2n + 2n + 2n…

Q: Let Calculate f(x) = ² Σ K=O k f(K) 2 k! where K=0,1₁.. X>0

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Q: Let E = {n E Z|n is even} and let O = {n E Z|n is odd}. Define a function f: E × O + Z by f(x, y) =…

A: Given: E=n∈ℤ|n is even O=n∈ℤ|n is odd f:E×O→ℤ such that fx,y=x+y To choose the correct option: The…

Q: Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]

A: Given if f∈Rα on a,b and g∈Rα on a,b. We have to prove that fg∈Rα on a,b. Since, f∈Rα on a,b then f…

Q: Consider following definition of function. f: X→X, f(x)≡(3x+11) mod 26, where X={0,1,2,....25}.…

A: f(x) = (3x+11) (mod 11) X = {0, 1, 2, ..., 25} GCD(3,26)=1

Q: Let f:Z→Z be defined by f(n) = 2n+ 1 for each n∈Z and g:R→R defined by g(x) = 2x+ 1 for each x∈R.…

A: Given function is Let f:Z→Z be defined by f(n) = 2n+ 1 for each n∈Z and g:R→R defined by g(x) = 2x+…

Q: Theorem 8.9: Let A ≤ R, let ƒ : A → R, and let r E A. The following are equivalent: 1. f is…

A: