Let X be a Set and X. A be a g-algebra on @Suppose AC X. show that the function f: XR given by flxx) = $-9, If X EA to, if x € AC is A-measurable if and only if A € A. When is a function S:X-R said to be Simple? © Let x = [0, ∞), at be the Borel 6-algebra On X and m be the Lebesgue measure on [0). Suppose that f: [0,00) +R is given by fr (x) = 20, for all x € [0, ∞). lim ff dm = flim fn dm? Justify your 74 answer. IS lim

Q: 4x-y<-1

A: Given  4x-y&lt;1 Can be rewritten as y&gt;4x-1 So we take the area above the line y = 4x -1 And all…

Q: 15. Let A be an infinite set. Prove that there exists an injective function f: N→ A.

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Q: Find example where of is continuous and an - bijective but f¹ is not

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Q: (c) h: Z → Z. h(x) = x³ (d) f: Z - Z, f(x) E - 4 (e) f: Z- Z, f(x) = 5x - 4

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Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A → A and ƒ: A → A. (a)…

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Q: Suppose f : A → A is a function and A has 7 elements. Prove each direction of the following…

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Q: (B1>0 (g) If a > b and c> d, then a + c > b + d. h) If a > b> 0 and c > d > 0, then ac > bd. i) If a…

A: For (f) ,We use a theorem that "if n∈ℤ and n≠0 then n2&gt;0.

Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: The solution of given problem is given as follows :

Q: b) A function g:Z→Z is defined by the rule g(m, n) = m + n Determine whether it is onto or not...

A: A function f:A→B is said to be onto or surjective if for all element b∈B there is an element a∈A…

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

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Q: b) A function g: Z → Zis defined by the rule g(m, n) = m + n Determine whether it is onto or not..

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

Q: C. Let T be defined onN N by (a, b)T (c, d) if and only if a c and b 2 d. Write down the full nature…

A: We will use the basic definition of reflexive, symmetric, antisymmetric and transitive realtion.

Q: Recall, a rule assigning elements of a set A to a set B is said to be well-defined if the assignment…

A: Given: is defined as .

Q: Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and…

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Q: Q. (a) Find an equation of the tangent plane to each surface at the given point. (b) Find symmetric…

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Q: Let f:X → Y be a linear function, where X and Y are normed spaces over the field of real numbers.…

A: The given linear function: Where X and Y are normed spaces over the field of Real numbers. To prove…

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

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Q: let a= {0,4,9,1.6} , b={0,3,6,12,20} and f a function. from a to b with f(x)=√x + x. A. Find f(0),…

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Q: For any subset A of R, define the function XA : R →R by if x E A 1, XA(") = 0, if x ¢ A. Let A and B…

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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: Prove that fo f = Idom f iff f-¹ is a function from ran f to dom f.

A: Prove That : f-1of=Idom iff f-1is a function from ran f to dom f.

Q: What is a affine function and what is it used to?  can you give a example

A: If y is proportional to x then we can write it as y∝x, we can change the proportionality sign by…

Q: The following functions all have domain {1, 2, 3, 4} and codomain {1, 2, 3, 4, 5}. For each,…

A: The domain of the given functions and the codomain .(a) (b) (c) The aim is to determine whether the…

Q: 12. Let Rs = {0, 1, 2, 3, 4}. (a) Define f: R5→ R5 by ƒ(x) = x² + 4 (mod 5) for all x e R5. Write…

A: We use the definition of function.

Q: Show that if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

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Q: Show that the function f: Real #s to Real #s defined by f(x)=x2 is not a homomorphism.

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Q: Let f = (1, 6) (2, 3, 5, 4). Compute each of the following powers of f.   f-2

A: We have given the permutation :  f=(1,6)(2,3,5,4) We have to compute : f-2

Q: Let T={[a, ) : a€R}, then T is not Borel algebra.

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Q: 3. Let ƒ : A → B be a function. Prove that if ƒ has an inverse, then fis bijective.

A: This question is related to topic function from real analysis.

Q: 13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all…

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Q: if f is measurable then - f is messlurable

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Q: Let X be the set {5, –6, 2}. For the first three parts of this problem you are asked to define a…

A: Consider the set 5,-6,2. The function is f:X→X and the relation is u~w⇔w=fu.   a) The relation is…

Q: f(x)=x2-1 g(x)=2x+1 prove if the following functions are onto prove if the following functions…

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Q: Let f : R → R and ECR. Prove f is continuous iff f-¹(E) is closed whenever ECR is closed.

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