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: 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…

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A:

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

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

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A:

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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=…

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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:

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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

A:

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: 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: