Let C([0, 1]) denote the set of all continuous functions f : [0, 1] → R, and let Qn (0, 1] = {r1,"2;, "3, . ..} be an enumeration of the set of rational numbers in [0, 1]. For f and g in C([0, 1]), define d(f, g) = 2lf(r:) – g(r:)|. i=0

Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

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Q: Consider the functions f : A → B, g : B → A and h : B → A such that if gf = 1 Ą and fh = 1B. Prove…

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Q: suppose g: A---> B and f: B---> C  are both one-to-one. prove f o g is one-to-one

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Q: Let Z be the set of integers, determine f:Z->Z, such that, for all integers a and b,…

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Q: Prove that the following is one- to one and onto:  f(x) =x3+2

A: To verify that the given function is one -one onto from R to R (R= the set of real numbers)

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: 11) Let H < K < G. Prove that |G:H|= |G:K| · |K:H| (Do not assume G is finite).

A: Given : H≤K≤G We have to prove that G : H=G : K. K : H

Q: Let A = {1,2,...,n} and consider the functions f: A → A. a. How many functions satisfy f(1) = 1? b.…

A: This question related to function and bijection.

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Q: Count the number X of functions y: (20, 21,..., 23) → (5,6,...,8) so that is not onto. X =

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Q: Let f.g.h:A - A be any bijective functions. Then (g oh on-

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Q: CONSIDER A FUNCTION OF R IN R DEFINED BY: x² + 1 X-2 SO, THE EXPRESSION: f(x) - f(3) x - 3 FOR 2 IS…

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Q: a) Give an example of an algebraic function f : Z → Z that is injective but not surjective. You do…

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Q: Let a and b be any real numbers such that a 0. Prove that a < b.

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Q: let G be the set of all triples of the form (k1, k2,1) or (k1,k2,-1), where the ki, i=1,2 are…

A: Given :  G = k1, k2, ±1 : k1, k2 ∈ℤ with the operation  k1, k2, 1l1, l2, -1 = k1+l2, k2+l1, -1 To…

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Q: Prove that, if a, b, and c are real numbers such that a <b< c, then |b| < max{Ja|, Jel}.

A: Consider the given: a, b ,c are real numbers  such that a&lt;b&lt;c

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Q: b) Prove or disprove: i. [x]-[x]. ii. For all positive real numbers x and y, [xy] ≤ [x] - [y].

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Q: Consider the set S = {x² + 1, 2x – 2, – x² + x}. The set S spans P2. If t(x) 2x? – 13x + 9 € P2 ,…

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Q: Consider the functions f : A → B, g : B → A and h : B → A such that if gf =14 and fh = 1B. Prove f…

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Q: 6. Based on what you know about the multiplicative principle, how many functions f:A- Bare there if…

A: Solution: