Consider E={-2+1/n} n=1 to infinity U (3,9) as a subset of R with the usual definition of < (less than) a

Q: Can you prove that it is injective by contrapositive? Also, how is f surjective when y exists in R?

A: Different equivalent definitions can be given for injectivity. The function f:A→B is injective if…

Q: 12. For each n € N, define A, = [0, 1+] and B₁ = [0,1]. - =1 (a) Prove that U1A,, is an interval,…

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Q: Suppose that f(x) is convex and continuous on an interval I. Show that dif(x1) + ·.+ A„f(xn) + Anf…

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Q: Prove that x² + ¹² +2111²2 = y 1112 where prime denotes differentiation w.r.t.s. 1 1+p¹² 4 p² o² + P

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Q: Q.1A) Evaluate 2 (4-x*) 4 cosy dydx 4-y

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Q: Let k be a positive integer. How many distinct functions are of the form g : [1..k] → [1..2k] such…

A: A function is defined when each value in the domain has exactly one value in the range. A…

Q: Juppose that the Mumber On ole det Bively by ogas for all nz2. Show that Oh= 2 +3( recu true for…

A: true

Q: 27. Let f(x)= [x²/3]. Find /(S) if a) S-2, 1, 0, 1, 2, 3). b) S= (0, 1, 2, 3, 4, 5). c) S = (1,5. 7.…

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Q: -. For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m-1.

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Q: Find the lexact and then appeximate) value (s) of K that will make the given function continous over…

A: Finding the left and right limits at x=4:

Q: Prove that there is an additive function T : R --&gt; R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

Q: Let g : B → N × N where B C Z given by g(m) = (|m| + 2, |m| + 3). Choose all subsets B such that g…

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Q: 2. Let N = {1,2,3,...}. Are the following functions injective, surjective and/or bijec- tive? (i) f:…

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Q: Find a ≤ b such that

A: ∫abx2-9dxApply the Sum Rule:…

Q: d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following…

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Q: A. If X~N(196,20), find x such that Pr(X < x) = 0.945.

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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: Prove f:N - N. F(X) = 5x is injoctive.

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Q: a_ if f(z) is analytic function, show th 2 8²) | f (2)| = + | f (2)| ² 78 Dy12

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Q: Which of the following function is NOT Q, (n)? А. п —333п? nlgn С. п-1gn D. 219 n E. None Е.

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Q: 2.59 Find || f |sup if a) f(x) = x on (-1, ). So if x E Q, b) f(x) = – x² .2 if x ¢ Q. -

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Q: mutually disjoint? Why or why not?

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Q: Find the term independent of of in the expansi Ansion of √x 3 + 3 222 to

A: Given expression is x3+32x210. The aim is to find the term independent in the expansion of the above…

Q: f. Is the domain of f(x.y)= In (6x + 5y“) bounded or unbounded? O Unbounded O Bounded

A: the domain will be Unbounded because

Q: Find ||f||sup if - f(x) = x on (-1,1). -. f(x) = 0 {²-2² if x EQ, -x² if x #Q, for all x ER.

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Q: which of the following statements is for t*t, the convolution oft with t" Select one: а. S. T(t –…

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Q: 3. (a) Find a unit in Z[√2] which is not equal to 1 or -1. Hint: Use 1 (b). (b) (*) Show that Z[√2]…

A: The objective of the question is two-fold. First, we need to find a unit in the ring Z[sqrt(2)] that…

Q: ) Let S be the non-empty set containing all a for which f(x): = monotonic for all xe R. Find S. 4a-7…

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Q: Let P(z) = z" +am-12"-1 + ... + a1z + ao be a polynomial of degree n 2 1 with a; # 0 for at %3D…

A: We will solve Q9 as mentioned. Given Pz=zn+an-1zn-1+…+a1z+a0 be a polynomial of degree n≥1 with ai≠0…

Q: Show that it is always true that f(AUB) = f(A) u f(B) and f(ANB) (f(A) AT (B), (@) show by example,…

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Q: let a and m be the integers with gcd(a,m)=1, what is the relation between ord m (a) and ord m (-a)

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Q: Let f be an entire function such that |f(2)| < A + B|z|" for all z E C, where A and B are positive…

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