Prove: If f € R(a) on [a, b] and if |f(x)| < M on [a, b] , then | S. f da| < M[a(b) – a(a)].

Q: Suppose f: R → R is the squaring function, that is, f(x) = x² for every real number . Mark all of…

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Q: 6. Determine the type of the following functions ( Injection, Surjection, Bijection?). a) Let f be…

A: The solution given as,   a:-  Let 'f' be the function from             a,b,c,d to1,2,3      and…

Q: Q: Interpret the vate of change f y = (-2 Jn=+1 + 5) (45 -2) with Yespect to

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Q: For a function f: A → B and subsets C and D of A, prove that f(CUD) = f(C) Uƒ(D).

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Q: For S in the domain of f, let f ( 5 ) = { f(x): * € S}. Let C and D f de subsets of the domain of f.…

A: S is in the domain of f fS=fx: x∈S Let C and D be subsets of the domain of f Then we have to show…

Q: (c) For any (a, y) E R?, let F(x, y) = (4x - 3y + 1, 3x +4y + 1). Then F is a similarity.

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Q: Suppose f: Q→ Z where (a) Is f one-to-one? Justify. ƒ(x) = [² 2 ¹]

A: One-one functions maps a image back to a single pre-image Onto function maps every element in…

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

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Q: Q. AfX-Y where X,Y € Z* , and 0<X <23 and, 0s Ys59, X is hour of clock and Y is minute of digital…

A: Given: f: X→Y , where X is hour of clock and Y is minute hand of clock It can be clearly observed…

Q: Suppose that f : A → B is a function, C C B and D C B. Prove that Š (C – D) = } (C) - } (D).

A: Solution :-

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: 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 on S is a function from S×S into S.

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Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

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Q: Theorem 10 Let (a, b] c R and f : [a, b] → R be a function of bounded variation on fa, b). Then f is…

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Q: Let ƒ be continuous and nonnegative on [a, b]. Show that there is c € [a, b] with f(c) = (±a få…

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Q: What is true about the function -- asx-m? OAM-f B) (x) 0 from below OC) f(x) is undefined OD)x) from…

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Q: Prove that f(x) = x ⋅ |x| is continuous at all points c in R

A: Given the function fx=x·x. We have to prove that fx=x·x is continuous for all points c in ℝ.

Q: Let f be the function defined by f (x) = x³ – x² – 6x. What is the absolute maximum value of f on…

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Q: (b) ƒ(AnB) ≤ f(A) Ñ ƒ(B)

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Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

Q: Consider the following statements 1- The function f defined by f(x) = x³ is decreasing in R 2- If a…

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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: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

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Q: 3. Let X = {1,2, 3, 4}, Y = {a, b, c} and let f: XY. Prove by contradic- tion that f can't have an…

A: Given that: X={1,2,3,4}and Y={a, b, c}and f:X→Y

Q: 1. Define function f : Z+ → Z such that f(n) = /n. (a) Is f one-to-one? Justify (prove) your answer.

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Q: ® let the functions: 4:R-A gR-R,with こXe2 Prove that: Tx(4g) =4Vxg +CVf)xg

A: Given two functions: f(x,y,z)=x2eyz3, g(x,y,z)=(x2z2,-xyz,y3ez) To prove  ∇×(fg)=f∇×g+(∇f)×g

Q: If f, g E R[a,b] with f < g on [a,b], prove that g.

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Q: 7. ) Let f: N→ Z with f(n)=n+4. Let A = {1,2,3} and B = {1 € Z \ I <10). Use these to find the…

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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: Prove: If f₁, f₂ € R(a) on [a, b] with f₁ ≤ f₂, then so f₁ da ≤ f f₂ da.

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Q: Given x = {5,10,15,20,25,30} and y function from X to Y. Let f(5) = 400, f(10) = 100, f(15) = 500, f…

A: x=5, 10, 15, 20, 25, 30  &amp;  y=100, 200, 300, 400, 500, 600. f5=400,  f10=100,  f15=500,…

Q: verify 'If yes, explain why. If no, provide a counterexample. the following statement. For every…

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Q: 8. (p f(A) ≤ f(B), then ACB. Let f: XY. Prove that for all A, B C X, if f is one-to-one and

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Q: Let f:2+ +2 be the function defined by if n>l00 f(f (nt 11)) othe rwise fln) =d n-10 what is the…

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Q: Complete the statement of the following defintion: Definition (Intermediate Value Property). A…

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Q: Show that f (a) = is Libschitz function on r+ 2

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Q: (8) Suppose that a is algebraic over F, and [F(a): F] = 7. What is [F(a): F?

A: The degree is computed in the next step.