Homework Answers
(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;).
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)
16 pts) PROBLEM 21. Let f:X →Y be a function, let Xi, X2SX andlet Yi, ½SY. ) Write down the definitions of f(Xi) and f (Y 。ín½) = f-'(%) nf-106). (ii) Prove that (ii) Prove that f(XinX)(xnf(...
16 pts) PROBLEM 21. Let f:X →Y be a function, let Xi, X2SX andlet Yi, ½SY. ) Write down the definitions of f(Xi) and f (Y 。ín½) = f-'(%) nf-106). (ii) Prove that (ii) Prove that f(XinX)(xnf(xa) (i) Find a counterexample to the statement (xinx) J(x)n(X) Do not show how you found the right ideas. Present detailed and carefully set out definitions and proofs only END of PROBLEM 21
16 pts) PROBLEM 21. Let f:X →Y be a function, let...
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
Problem 6.8. Let X = {1, 2, 3}, Y = {a, b, c, d, e}. (a)...
Problem 6.8. Let X = {1, 2, 3}, Y = {a, b, c, d, e}. (a) Let f : X → Y be a function, given by f(1) = a, f(2) = b, f(3) = c. Prove there exists a function g : Y → X such that g ◦f = id X . Is g the inverse function to f? (Hint: define g on f(X) to make g ◦ f = id X . Then define g on Y...
4. Let X and Y be any sets and let F be any one-to-one (injective) function...
4. Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y. Prove that for every subset A CX: (a) (10 points) AC F-(F(A)). (b) (10 points) F-1(F(A)) C A.
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z)...
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
how do u do 6? F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f...
how do u do 6?
F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f is defined on a set of real numbers. Determine whether or not f is one-to-one and justify your answers. (a) f(x) = **!, for all real numbers x #0 (6) f(x) = x, for all real numbers x (c) f(x) = 3x=!, for all real numbers x 70 (d) f(x) = **, for all real numbers x 1 (e) f(x) = for all real...
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2....
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
(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;).
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)
16 pts) PROBLEM 21. Let f:X →Y be a function, let Xi, X2SX andlet Yi, ½SY. ) Write down the definitions of f(Xi) and f (Y 。ín½) = f-'(%) nf-106). (ii) Prove that (ii) Prove that f(XinX)(xnf(xa) (i) Find a counterexample to the statement (xinx) J(x)n(X) Do not show how you found the right ideas. Present detailed and carefully set out definitions and proofs only END of PROBLEM 21
16 pts) PROBLEM 21. Let f:X →Y be a function, let...
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
4. Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y. Prove that for every subset A CX: (a) (10 points) AC F-(F(A)). (b) (10 points) F-1(F(A)) C A.
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
how do u do 6?
F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f is defined on a set of real numbers. Determine whether or not f is one-to-one and justify your answers. (a) f(x) = **!, for all real numbers x #0 (6) f(x) = x, for all real numbers x (c) f(x) = 3x=!, for all real numbers x 70 (d) f(x) = **, for all real numbers x 1 (e) f(x) = for all real...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
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