Homework Answers
2. For i = 1, ..., k, let Tį be open subsets of Rņi and let...
2. For i = 1, ..., k, let Tį be open subsets of Rņi and let pi : Ti → Rři be C1 transformations such that each pi is a Cl bijection onto its image with C1 inverse. If each pi(T;) and ITX=1 Pi(T;) are measurable, prove that Vor" (TIP:() = [ Vol" (0:17:)) where m= ni t... + nk
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3}...
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets...
8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets of X and is a measure sequenc o clemenis of K We delin lim supn(Fn) liminfn(En)- U then prove: (a) lim in(E)) lim inf(u(E,) (b) T J (c) If sum E,)x, then (lim sup(E)) = 0 x X) <oc lor somc nE N, then lim supn (Fn)> lim sup(u(F,n ))
8 arbitrary set. K is Cousider...
1. Let S and T be subsets of the universal set U. Use the Venn diagram...
1. Let S and T be subsets of the universal set
U. Use the Venn diagram on the right and the given data below to
determine the number of elements in each basic region.
n(U)=25
n(S)=13
n(T)=14
n(SUT)=19
Region I contains _____ elements
Region II contains _____ elements
Region III contains _____ elements
Region IV contains _____ elements
______________________________________________________________________________________________________
2. Let R, S, and T be subsets of the universal
set U. Use the Venn diagram on the right and...
OTO (7) (a) Let T = (a1, ..., ak) be a k-cycle in Sn, and let...
OTO (7) (a) Let T = (a1, ..., ak) be a k-cycle in Sn, and let o E Sn. Prove that is the k-cycle (o(a), o(az),..., 0(ak)) (b) Let o,t e Sn. Prove that if t is a product of r pairwise disjoint cycles of lengths k1,..., kr, respectively, where kit..., +kr = n, then oto-1 is also a product of r pairwise disjoint cycles of lengths k1,..., kr. (c) Let T1 and T2 be permutations in Sn. Prove that...
Please provide the theorems and definitions you use. 1. Let K be a subgroup of a...
Please provide the theorems and definitions you use.
1. Let K be a subgroup of a group G. Let T denote the set of all distinct right cosets of K in G and A(T) be the permutation group of T. Prove the following statements. (a) For each a EG, the function fa:T T given by fa(Kb) = Kba is a bijection. (b) The function : G + A(T) given by pla) = fa-1 is a group homomorphism whose kernel is...
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin,...
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let...
Topology (a) For each subset A of NV0), define eA є loo such that the k-th component of eA is 1 if k є A and 0 otherwise. Define B-(Bde (eA; 1/2) : A N\ {0)). Recall I. (i) If AメB are subsets of N \...
Topology
(a) For each subset A of NV0), define eA є loo such that the k-th component of eA is 1 if k є A and 0 otherwise. Define B-(Bde (eA; 1/2) : A N\ {0)). Recall I. (i) If AメB are subsets of N \ {0), find the value of doc (eA, eB). (ii) Show that B is a collection of disjoint open balls in 100. iii) By quoting relevant results, justify whether or not the collection B is...
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) P...
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...
2. For i = 1, ..., k, let Tį be open subsets of Rņi and let pi : Ti → Rři be C1 transformations such that each pi is a Cl bijection onto its image with C1 inverse. If each pi(T;) and ITX=1 Pi(T;) are measurable, prove that Vor" (TIP:() = [ Vol" (0:17:)) where m= ni t... + nk
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets of X and is a measure sequenc o clemenis of K We delin lim supn(Fn) liminfn(En)- U then prove: (a) lim in(E)) lim inf(u(E,) (b) T J (c) If sum E,)x, then (lim sup(E)) = 0 x X) <oc lor somc nE N, then lim supn (Fn)> lim sup(u(F,n ))
8 arbitrary set. K is Cousider...
1. Let S and T be subsets of the universal set
U. Use the Venn diagram on the right and the given data below to
determine the number of elements in each basic region.
n(U)=25
n(S)=13
n(T)=14
n(SUT)=19
Region I contains _____ elements
Region II contains _____ elements
Region III contains _____ elements
Region IV contains _____ elements
______________________________________________________________________________________________________
2. Let R, S, and T be subsets of the universal
set U. Use the Venn diagram on the right and...
OTO (7) (a) Let T = (a1, ..., ak) be a k-cycle in Sn, and let o E Sn. Prove that is the k-cycle (o(a), o(az),..., 0(ak)) (b) Let o,t e Sn. Prove that if t is a product of r pairwise disjoint cycles of lengths k1,..., kr, respectively, where kit..., +kr = n, then oto-1 is also a product of r pairwise disjoint cycles of lengths k1,..., kr. (c) Let T1 and T2 be permutations in Sn. Prove that...
Please provide the theorems and definitions you use.
1. Let K be a subgroup of a group G. Let T denote the set of all distinct right cosets of K in G and A(T) be the permutation group of T. Prove the following statements. (a) For each a EG, the function fa:T T given by fa(Kb) = Kba is a bijection. (b) The function : G + A(T) given by pla) = fa-1 is a group homomorphism whose kernel is...
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let...
Topology
(a) For each subset A of NV0), define eA є loo such that the k-th component of eA is 1 if k є A and 0 otherwise. Define B-(Bde (eA; 1/2) : A N\ {0)). Recall I. (i) If AメB are subsets of N \ {0), find the value of doc (eA, eB). (ii) Show that B is a collection of disjoint open balls in 100. iii) By quoting relevant results, justify whether or not the collection B is...
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...
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