Real Analysis
Let X be a non- empty set and define,:
d(x,y)=
Show that d1 and d2 are equivalent metrics on X.
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Have to get an idea of how i am doing on this problem. Whould be nice...
Have to get an idea of how i am doing on this problem. Whould be
nice to get a good explaination for each part of the problem. d1
and d2 is the two different metrics, p ,Y.
Problem 2. Consider first the following definition: Definition. Let X be a set and let pand be two metrics on X. We say that p and are equivalent if the open balls in (X, p) and (x,y) are "nested". More precisely, p and...
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...
(5) Here is a fascinating equivalence for being a complete metric space that we will use...
(5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space. (b) ** (10 points) Show that the following are equivalent: • (X, d) is complete; • for every family of non-empty closed subsets Fo, F1, F2, ... of X such that F, 2 F12 F22... and limn700 diam( Fn) = 0, it holds that Nnen Fn = {a} for some a € X. (Hint: for the reverse...
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are...
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are said to be uniformly equivalent if the identity map from (X, d) to (X, d) a nd its in- verse are uniformly continuous. (a) Prove that uniform equivalence is indeed an equivalence relation on the class of metrics on X. (b) Let (X,d) and (x, ) be uniformly equivalent. Are the following true or false? (i) If (X, d) is bounded, then must also...
i) Does Lebesgue lemma hold true in the plane? Justify your answer! ii) Let (X, d1) be a compact metric space and (Y, d2) a metric space. Suppose that f : X → Y is continuous. Use Lebesgue lemma to sh...
i) Does Lebesgue lemma hold true in the plane? Justify your answer! ii) Let (X, d1) be a compact metric space and (Y, d2) a metric space. Suppose that f : X → Y is continuous. Use Lebesgue lemma to show that for every > 0 there exists δ > 0 such that if d1(x, y) < δ then d2(f(x), f(y)) < , that is, f is uniformly continuous.
Let X be a non-empty set. Show that the only dense subset of X with respect...
Let X be a non-empty set. Show that the only dense subset of X with respect to the discrete metric ddise is X. The whole set of any metric spaces is always dense, so this question is really asking you to exclude all other possibilities. Show that if (X, d) is a metric space and has dense subset A + X, then (X, d) is not topologically equivalent to (X, ddisc). (Note that this is another way of showing that...
Let P(X) be the power set of a non-empty set X. For any two subsets A...
Let P(X) be the power set of a non-empty set X. For any two subsets A and B of X, define the relation A B on P(X) to mean that A union B = 0 (the empty set). Justify your answer to each of the following? Isreflexive? Explain. Issymmetric? Explain. Istransitive? Explain.
b) Let A-2,4 and B 1,3,5). Define the nonempty and pairwise different relations U, V, W...
b) Let A-2,4 and B 1,3,5). Define the nonempty and pairwise different relations U, V, W C A x B as follows: . (x,y) E U implies 7 .(x,y) E V implies r > y . z > y implies (z, yje W. i. For wich of the above tasks would the empty set be a valid solution (if the "non- empty" wasn't given in the task)? ii. Determine if the relations U, V and W are functions and reason...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
(2 pts each) Find a different equivalent form of the statements. Justify your answers using Laws...
(2 pts each) Find a different equivalent form of the statements. Justify your answers using Laws of equivalence or otherwise. (a) Not all men are Scientists. (b) If you are a computer science major you will need discrete mathematics. (10 pts) Let R be an equivalence relation on a non empty set X. So R C X X X. R is reflexive: Vx E X, (x,x) E R, R is symmetric: Vx, y E X, (x,y) E R = (y,x)...
Have to get an idea of how i am doing on this problem. Whould be
nice to get a good explaination for each part of the problem. d1
and d2 is the two different metrics, p ,Y.
Problem 2. Consider first the following definition: Definition. Let X be a set and let pand be two metrics on X. We say that p and are equivalent if the open balls in (X, p) and (x,y) are "nested". More precisely, p and...
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...
(5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space. (b) ** (10 points) Show that the following are equivalent: • (X, d) is complete; • for every family of non-empty closed subsets Fo, F1, F2, ... of X such that F, 2 F12 F22... and limn700 diam( Fn) = 0, it holds that Nnen Fn = {a} for some a € X. (Hint: for the reverse...
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are said to be uniformly equivalent if the identity map from (X, d) to (X, d) a nd its in- verse are uniformly continuous. (a) Prove that uniform equivalence is indeed an equivalence relation on the class of metrics on X. (b) Let (X,d) and (x, ) be uniformly equivalent. Are the following true or false? (i) If (X, d) is bounded, then must also...
Let X be a non-empty set. Show that the only dense subset of X with respect to the discrete metric ddise is X. The whole set of any metric spaces is always dense, so this question is really asking you to exclude all other possibilities. Show that if (X, d) is a metric space and has dense subset A + X, then (X, d) is not topologically equivalent to (X, ddisc). (Note that this is another way of showing that...
b) Let A-2,4 and B 1,3,5). Define the nonempty and pairwise different relations U, V, W C A x B as follows: . (x,y) E U implies 7 .(x,y) E V implies r > y . z > y implies (z, yje W. i. For wich of the above tasks would the empty set be a valid solution (if the "non- empty" wasn't given in the task)? ii. Determine if the relations U, V and W are functions and reason...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
(2 pts each) Find a different equivalent form of the statements. Justify your answers using Laws of equivalence or otherwise. (a) Not all men are Scientists. (b) If you are a computer science major you will need discrete mathematics. (10 pts) Let R be an equivalence relation on a non empty set X. So R C X X X. R is reflexive: Vx E X, (x,x) E R, R is symmetric: Vx, y E X, (x,y) E R = (y,x)...
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