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Exercise 2.2. Give an alternative description of the sets specified below. 1. {DER : x2 =...
Question 1: Let R be the set of real numbers and let 2R be the set...
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by...
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by To(x) = 0 VxEFN. (i) What is R(To)? (ii) Is To onto? (iii) What is N(To)? (iv) Is To one-to-one? (v) What is (To]s? b) The identity transformation. We define the identity transformation, Tj: Fn + En by Ty(x) = x V xEFN. (i) What is R(Ti)? (ii) Is T, onto? (iii) What is N(T)? (iv) Is T one-to-one? (v) What is Ti]s? Question...
Prove that a disjoint union of any finite set and any countably infinite set is countably...
Prove that a disjoint union of any finite set and any countably infinite set is countably infinite. Proof: Suppose A is any finite set, B is any countably infinite set, and A and B are disjoint. By definition of disjoint, A ∩ B = ∅ Then h is one-to-one because f and g are one-to one and A ∩ B = 0. Further, h is onto because f and g are onto and given any element x in A ∪...
Exercise 6.6.1. Let f a, bR be a differentiable function of one variable such that If,(x) 1 for a...
Please prove in detail (Exercise 6.6.1):
Exercise 6.6.1. Let f a, bR be a differentiable function of one variable such that If,(x) 1 for all x є [a,b]. Prove that f is a contraction. (Hint: use the mean-value theorem, Corollary 10.2.9.) If in addition |f'(x)| <1 for all x [a,b] and f, is continuous, show that f is a strict contraction.
Exercise 6.6.1. Let f a, bR be a differentiable function of one variable such that If,(x) 1 for all...
Exercise 1. This exercise builds on the method used to prove that if a function differetiable at ...
#1 & #2
Exercise 1. This exercise builds on the method used to prove that if a function differetiable at a point b, then it is also continuous at b. Suppose g : (-1,1) → R is a function such that g(0) = 7 and lim 9)-7-10 exists. Define G())7-10 on-l < x < 1 when x need to know the value of λ, but its existence is necessary in what follows. 0. Let λ be the limit of G(x)...
Exercise 1. Tangent plane (15 pts) Let (5) be the surface given by the following equation....
Exercise 1. Tangent plane (15 pts) Let (5) be the surface given by the following equation. x2+y2 = 1+z2 An equation of the tangent plane to (S) at A(1,2,2) is: a. 2x + 4y – 4z = 1 b. x + y - z=0 c. x + 2y – 2z = 1 d. x + y - z = 2 e. None of the above a. b. C. O d. e. Exercise 2. Directional derivative (6 pts + 9 pts)...
Exercise 7.H. 7.Н. Show that every number in the Cantor set has a ternary (-base 3)...
Exercise 7.H.
7.Н. Show that every number in the Cantor set has a ternary (-base 3) expan- sion using only the digits 0, 2 7.I. Show that the collection of "right hand" end points in F is denumerable. Show that if all these end points are deleted from F, then what remains can be put onto one-one correspondence with all of [0, 1). Conclude that the set F is not
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a,...
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
e. 1 Puestion 16 Let E be the solid tetrahedron with vertices (1,1,0), (1,0,4), (0,1,4), (1,1,4)....
e. 1 Puestion 16 Let E be the solid tetrahedron with vertices (1,1,0), (1,0,4), (0,1,4), (1,1,4). Let D Hot yet answered be the projection of E onto xy-plane. If I dV = / f(x,y) dĄ, then f(x, y) = Marked out of 3.00 Flag question Select one: O a. 3 + 3x + 3y b. 1 + 5x + 5y c. 1 + 6x + 6y d. None of these e. 2 + 2x + 2y O f. 4 +...
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by To(x) = 0 VxEFN. (i) What is R(To)? (ii) Is To onto? (iii) What is N(To)? (iv) Is To one-to-one? (v) What is (To]s? b) The identity transformation. We define the identity transformation, Tj: Fn + En by Ty(x) = x V xEFN. (i) What is R(Ti)? (ii) Is T, onto? (iii) What is N(T)? (iv) Is T one-to-one? (v) What is Ti]s? Question...
Please prove in detail (Exercise 6.6.1):
Exercise 6.6.1. Let f a, bR be a differentiable function of one variable such that If,(x) 1 for all x є [a,b]. Prove that f is a contraction. (Hint: use the mean-value theorem, Corollary 10.2.9.) If in addition |f'(x)| <1 for all x [a,b] and f, is continuous, show that f is a strict contraction.
Exercise 6.6.1. Let f a, bR be a differentiable function of one variable such that If,(x) 1 for all...
#1 & #2
Exercise 1. This exercise builds on the method used to prove that if a function differetiable at a point b, then it is also continuous at b. Suppose g : (-1,1) → R is a function such that g(0) = 7 and lim 9)-7-10 exists. Define G())7-10 on-l < x < 1 when x need to know the value of λ, but its existence is necessary in what follows. 0. Let λ be the limit of G(x)...
Exercise 1. Tangent plane (15 pts) Let (5) be the surface given by the following equation. x2+y2 = 1+z2 An equation of the tangent plane to (S) at A(1,2,2) is: a. 2x + 4y – 4z = 1 b. x + y - z=0 c. x + 2y – 2z = 1 d. x + y - z = 2 e. None of the above a. b. C. O d. e. Exercise 2. Directional derivative (6 pts + 9 pts)...
Exercise 7.H.
7.Н. Show that every number in the Cantor set has a ternary (-base 3) expan- sion using only the digits 0, 2 7.I. Show that the collection of "right hand" end points in F is denumerable. Show that if all these end points are deleted from F, then what remains can be put onto one-one correspondence with all of [0, 1). Conclude that the set F is not
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
e. 1 Puestion 16 Let E be the solid tetrahedron with vertices (1,1,0), (1,0,4), (0,1,4), (1,1,4). Let D Hot yet answered be the projection of E onto xy-plane. If I dV = / f(x,y) dĄ, then f(x, y) = Marked out of 3.00 Flag question Select one: O a. 3 + 3x + 3y b. 1 + 5x + 5y c. 1 + 6x + 6y d. None of these e. 2 + 2x + 2y O f. 4 +...
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