Task 4 Let and X={x1,.., xn}µbe a mapping defined on F: =2* as follows: for 0, H(A)= A=Ø k for #(A)=k n' Show that u it is a finite measure on the space (X,F).

Q: 6. (i) Let f: (X, J) → (Y, U), where (X, T) and (Y, U) are two topolog- ical spaces. Show that if…

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Q: Sketch a contour map for f(z. y) using the k-values, k = 1, 2, 3. |f(x, y) = |x| + sin(y)

A: Given : f(x,y)=|x|+sin(y)   To sketch : contour using the k-values k=1,2,3

Q: 1 ExL =5 (ex-5) f(x). g(»)dx (F-9) ニ Let v be the vector space of oll continous real valued function…

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Q: Find L{f (t)}using the First Shifting Theorem, Note: a, b, e, w, k and n are constants 1. Il…

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Q: is The functionFIO-Vx4iscontinusus onR. It cosă

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Q: sin z f(2) = z³(z² + 2z + 2)

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Q: Let f:R' -R' be a mapping given by f(x,y.z) = (u(x,y, z), v(x, y.z), w(x, y. z)). Let g: R -R' be…

A: We have ,  f:R3→R3 , fx,y,z=ux,y,z,vx,y,z,wx,y,zg:R3→R3 , gu,v,w=u-w,v+w,u+wAnd , h=gof⇒hx,y,z=g…

Q: Prove that if f:X + Y is a one-to-one and onto function with inverse function f- : Y → X, then fof-l…

A: Given that, f:X→Y is a one-to-one and onto function with inverse function f-1:Y→X

Q: (6) Are all continuous functions on ID differentiable on D?

A: we have to identify the given statement is true or not

Q: Evaluate using the result of Exercise 54

A: We use the half angle formula

Q: Let f: X → Y and g: Y → V be functions. Show the following: (a) f is injective 3h : Y→ X such that…

A: Given Data: Let f:X→Y and g : Y→V be functions.To Show that,(a) f is injective ⇔∃h :Y→X  such that h…

Q: Let f [0, 6] → R be the function defined as : f(x) = = -2, [4/x], if 0 < x < 1, if 1 ≤ x ≤ 6. Show…

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Q: Evaluate f. F.dr where F(x, y, z) = xyi + (x + y)j + x²k along the line segment from: A (1,3,1) to B…

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Q: Let f function from (R,d) to (R,d ) defined as f(x) = x , where d is .the discrete metric then f is…

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Q: Let X be f(x) = x^(2/3) , X is a subset of R x R satisfying the given equation. Define a bijective…

A: Given X be the set of all points x,fx, where fx=x23. That is X=x,x23: x∈ℝ. Now define a map g:X→ℝ as…

Q: Theorem 1 (For open convex domain). Suppose 2 is a convex open set. Let f be analytic in 2 \{a} and…

A: We can break the integrand {\displaystyle f}, as well as the differential {\displaystyle dz}…

Q: Sketch the graph of a function that is continuous on [1, 5] and has the given properties. A y 3.0…

A: Absolute maximum point is the point at where the function obtains its greatest possible value.…

Q: tch the following contour diagrams (cross-sections to the z-axis) to the following fund II) IV) 0.5…

A: Given contour diagrams and their functions. The objective is to match the contour diagrams with…

Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

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

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Q: Let (f) be a continuous function with the definition area a closed interval. Show that if (f) has an…

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Q: 3. Answer two of the following three questions: (a) Let f: R→ R be a function defined by f(x) = Is f…

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Q: f (x, y) of a smooth real-valued function f defined on a bounded Consider the graph z subset D of…

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Q: _x?yesin (xy2) x2+y2 -Let a function g be defined on R2 at g (x, y) Use: g(x,0), g (x, kx), g (x,…

A: We g(X,y ) first

Q: Prove that A = {x : x € X and ƒÃ(x) = 0}. (iii) Suppose A and B are nonempty disjoint closed subsets…

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Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

A: we need to find those points where the denominator ( z^4+1) equals to 0

Q: A contour map is shown for a function f on the square R= [0,8] x [0,8]. Assume a = 4, b = 8, c = 4,…

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Q: What are the domain and range of f? What is the inverse of f? Corppute fof fof. 1.4. REFERENCES AND…

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Q: 2. Given f (x, y) = x + 2y, find F(x, y) =Vf and sketch it along with the contour map of f.

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Q: Determine the set of points domains at which the function is continuous.  G(x,y)= 2x tan y

A: To Find: Set of points in domain at which the function 2x tany is continuous.

Q: Let f be continuous over [a,b] and differentiable over (a,b). Then there exists at least one point…

A: Let f be continuous over a,b and differentiable over a,b. Then there exits at leastone point c∈a,b…

Q: Let S' denote the set of (x, y) E R × R such that x? + y² This space is called the circle. Define a…

A: S1={(x,y)∈ℝ2/x2+y2=1} and p:ℝ→S1 by pt=cos2πt,sin2πt. To show: p is continuous, open,closed.