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…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

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

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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: Which of the following are continuous and piecewise polynomials with respect to the nodes o, x1, x2,…

A: Continuous and pocewise continuous of the polynomial

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: 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.

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

A: Given function φ:(20,21,...,23)→(5,6,...,8)

Q: Let a, b, c. and d be any real numbers such that a < b and c < d. Prove that [a, b] is equivalent to…

A: Answer

Q: Show that if x,y,z,w are real numbers such that x < y and z ≤w, then x + z < y + w.

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

A: Since this is a multiple question according to the Bartleby Answering rule, only first question is…

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

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Q: A function f:R →→R, where R is the set of real- ax² + 5x - 8 numbers, is defined by f(x) = Find a +…

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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

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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