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Exercise 1.3. Let {A,:y er be a family of sets indexed by ſ. a) Show that...
For the indexed family of sets D={Dn : n ∈ N}
For the indexed family of sets D={Dn : n ∈ N}, where Dn =(2 - 1/n, 4 + 1/n) for n ∈ N, find a. The union of the family D b. The intersection of the family D
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI 5. Let f: X → Y. Pr...
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI
(3 pts each) For each of the following find an indexed collection {An}nen of distinct sets...
(3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n=1 An = {0} (b) Un=1 An = [0, 1] (c) n=1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category with properly written domain D and range R such that (a) There exists a subset S of D with f-'[F(S)] + S (b) There exists...
2. Let A = {Aq: a € A} be a family of sets and let B...
2. Let A = {Aq: a € A} be a family of sets and let B be a set. Prove that (a) Bn UA=U (BOA). αΕΔ QE A (b) Let 4 = {Aq: A E A} and let B = {Beß ef}. Use (a) to write (4) (Uda) (UB) UBR as a union of intersections.
Exercise 425 Let k and n be positive integers, let v eR”, and let A €...
Exercise 425 Let k and n be positive integers, let v eR”, and let A € Mkxn(R). Show that Av = 0 if and only if A? Av= 0.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show tha...
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show that the rank of A must be 2.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a)...
Exercise 9. Let n 2 2 be a positive integer. Let a -(ri,...,^n) ER". For any...
Exercise 9. Let n 2 2 be a positive integer. Let a -(ri,...,^n) ER". For any a,y E R" sphere of radius 1 centered at the origin. Let x E Sn-be fixed. Let v be a random vector that is uniformly distributed in S"1. Prove: 10Vn
(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;).
Let A = {1, 2, 3} and B = {2, 3, 4, 5}. Find the cardinalities...
Let A = {1, 2, 3} and B = {2, 3, 4, 5}. Find the cardinalities of the following sets: (i) A ∪ B (ii) A ∩ B (iii) A \ B (iv) B \ A (v) P(A ∪ B) Exercise 1.2. Let A = {◦, {◦}, {∅}} and let B = {∅, {◦}}. Find the cardinalities of the following sets: (i) A ∪ B (ii) A ∩ B (iii) A \ B (iv) A × B (v) P(A) Exercise...
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C) (5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x...
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI
(3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n=1 An = {0} (b) Un=1 An = [0, 1] (c) n=1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category with properly written domain D and range R such that (a) There exists a subset S of D with f-'[F(S)] + S (b) There exists...
2. Let A = {Aq: a € A} be a family of sets and let B be a set. Prove that (a) Bn UA=U (BOA). αΕΔ QE A (b) Let 4 = {Aq: A E A} and let B = {Beß ef}. Use (a) to write (4) (Uda) (UB) UBR as a union of intersections.
Exercise 425 Let k and n be positive integers, let v eR”, and let A € Mkxn(R). Show that Av = 0 if and only if A? Av= 0.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show that the rank of A must be 2.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a)...
Exercise 9. Let n 2 2 be a positive integer. Let a -(ri,...,^n) ER". For any a,y E R" sphere of radius 1 centered at the origin. Let x E Sn-be fixed. Let v be a random vector that is uniformly distributed in S"1. Prove: 10Vn
(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;).
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
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