Homework Answers
6. Let f:A B be a function with domain A and codomain B. Let S and...
6. Let f:A B be a function with domain A and codomain B. Let S and T be subsets of the domain A a) Prove: f(ST)cf(S)n f(T) b) Give an example to show it is possible that f(SOT) f(S)nf (T). Name the domain, codomain, function, and sets S and T c) Let U and V be subsets of the codomain B. Prove: f (Unv)= f"(U)nfV)
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except
(6)...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except
(6)...
(1) Let X and Y be sets. Let f be a function from X to Y,...
(1) Let X and Y be sets. Let f be a function from X to Y, (a) IF BEY, recall that F-'(B) = {xeX \flyeBX(y,x) ef-)}. Prove that f'(B)={xeX | fk)e B}. (hint: Reprember that even though t is a thought is a function, the relation f may well not be itself a function.) Al b) Let {B; \je J} be an inbred family of subsets of Y. Prove that of "b) = f'(21B;).
Please explain why each answer is wrong or correct Thanks 15) Let S be an infinite...
Please explain why each answer
is wrong or correct
Thanks
15) Let S be an infinite and let T be a countably infinite set. Let S be the complement of S. If S and T are both subsets of real numbers, which of the following pairs of sets must be of the same cardinality? a) T, SOT b) S, SUT c) T, SUT d) Both A and B e) Both A and C f) None of these
left f:A->B and let D1, D2, and D be subsets of B prove or disprove f^-1(D1UD2)=f^-1(D1)Uf^-1(D2)...
left f:A->B and let D1, D2, and D be subsets of
B
prove or disprove
f^-1(D1UD2)=f^-1(D1)Uf^-1(D2)
does the proof change when it says subset of B vs
subset of A
let f:A->B and let D1, D2, and D be subsets of A. Prove or Disprove F^-1(D1UD2)=F^-1 (D1)UF^-1(D2)
5pt 1. Let g() = | f(t) dt, where f is the function whose graph is...
5pt 1. Let g() = | f(t) dt, where f is the function whose graph is shown below on the interval [0, 5). The graph consists of two straight line segments. - - - ------ -1- - - - - - --1- - -1- - - - - - - (a) Find g(1) and g(3). (b) On what interval(s) is g(x) decreasing? (c) At what x-value(s) in (0,5) does the local maximum of g occur? (d) At what x-value(s) in...
Let f : R2 → R be a uniformly continuous function and assume that If(y,t)| M. Let yo E R. The goal of this exercise...
Let f : R2 → R be a uniformly continuous function and assume that If(y,t)| M. Let yo E R. The goal of this exercise is to show the existence of a function φ : [0, 1] → R that solves the initial value problem o'(t)-F(d(t),t), ф(0)-Yo (a) Show that there is a function n1,R that satisfies t <0 n(リーレ0+.GF(du(s-1/n),s)ds, t20. Hint: Define фп first on [-1,0] , then define фп。n [0,1 /n), then on [1/n, 2/n], and so on...
Where Let n(t) be a fixed strictly positive continuous function on (a, b). define H, =...
Where
Let n(t) be a fixed strictly positive continuous function on (a, b). define H, = L([a,b], 7) to be the space of all measurable functions f on (a, b) such that \n(t)dt <0. Define the inner product on H, by (5,9)n = [ f(0)9€)n(t)dt (a) Show that H, is a Hilbert space, and that the mapping U:f →nif gives a unitary correspondence between H, and the usual space L-([a, b]). We were unable to transcribe this image
T F 5, Σ = {a,b), L = { s: s = anbm, nzn, m20, Isl s IP(Σ)13. (Th not longer than the number of elements in the power set of 2.) The re language pumping theorem could show that L RLs. T F 6. An N...
T F 5, Σ = {a,b), L = { s: s = anbm, nzn, m20, Isl s IP(Σ)13. (Th not longer than the number of elements in the power set of 2.) The re language pumping theorem could show that L RLs. T F 6. An NDFSM that recognizes a language L may have computation branc at is, s e L iff s is it accepts a string w L. 7. ISI = Ko, where S is a set. Ir(s)l...
6. Let f:A B be a function with domain A and codomain B. Let S and T be subsets of the domain A a) Prove: f(ST)cf(S)n f(T) b) Give an example to show it is possible that f(SOT) f(S)nf (T). Name the domain, codomain, function, and sets S and T c) Let U and V be subsets of the codomain B. Prove: f (Unv)= f"(U)nfV)
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except
(6)...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except
(6)...
(1) Let X and Y be sets. Let f be a function from X to Y, (a) IF BEY, recall that F-'(B) = {xeX \flyeBX(y,x) ef-)}. Prove that f'(B)={xeX | fk)e B}. (hint: Reprember that even though t is a thought is a function, the relation f may well not be itself a function.) Al b) Let {B; \je J} be an inbred family of subsets of Y. Prove that of "b) = f'(21B;).
Please explain why each answer
is wrong or correct
Thanks
15) Let S be an infinite and let T be a countably infinite set. Let S be the complement of S. If S and T are both subsets of real numbers, which of the following pairs of sets must be of the same cardinality? a) T, SOT b) S, SUT c) T, SUT d) Both A and B e) Both A and C f) None of these
left f:A->B and let D1, D2, and D be subsets of
B
prove or disprove
f^-1(D1UD2)=f^-1(D1)Uf^-1(D2)
does the proof change when it says subset of B vs
subset of A
let f:A->B and let D1, D2, and D be subsets of A. Prove or Disprove F^-1(D1UD2)=F^-1 (D1)UF^-1(D2)
5pt 1. Let g() = | f(t) dt, where f is the function whose graph is shown below on the interval [0, 5). The graph consists of two straight line segments. - - - ------ -1- - - - - - --1- - -1- - - - - - - (a) Find g(1) and g(3). (b) On what interval(s) is g(x) decreasing? (c) At what x-value(s) in (0,5) does the local maximum of g occur? (d) At what x-value(s) in...
Let f : R2 → R be a uniformly continuous function and assume that If(y,t)| M. Let yo E R. The goal of this exercise is to show the existence of a function φ : [0, 1] → R that solves the initial value problem o'(t)-F(d(t),t), ф(0)-Yo (a) Show that there is a function n1,R that satisfies t <0 n(リーレ0+.GF(du(s-1/n),s)ds, t20. Hint: Define фп first on [-1,0] , then define фп。n [0,1 /n), then on [1/n, 2/n], and so on...
Where
Let n(t) be a fixed strictly positive continuous function on (a, b). define H, = L([a,b], 7) to be the space of all measurable functions f on (a, b) such that \n(t)dt <0. Define the inner product on H, by (5,9)n = [ f(0)9€)n(t)dt (a) Show that H, is a Hilbert space, and that the mapping U:f →nif gives a unitary correspondence between H, and the usual space L-([a, b]). We were unable to transcribe this image
T F 5, Σ = {a,b), L = { s: s = anbm, nzn, m20, Isl s IP(Σ)13. (Th not longer than the number of elements in the power set of 2.) The re language pumping theorem could show that L RLs. T F 6. An NDFSM that recognizes a language L may have computation branc at is, s e L iff s is it accepts a string w L. 7. ISI = Ko, where S is a set. Ir(s)l...
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