a set (any set of objects) is said to be countable if it is either finite...
a set (any set of objects) is said to be countable if it is either finite or there is an enumeration (list) of the set. show that the following properties hold for arbitrary countable sets:
a) All subsets of countable sets are countable
b) any union of a pair of countable sets is countable
c) all finite sets are countable
Homework Answers
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a set (any set
of objects) is said to be countable if it is either finite...
Identify the correct steps involved in proving that the union of a countable number of countable...
Identify the correct steps involved in proving that the union of a countable number of countable sets is countable. (Check all that apply.) Check All That Apply Since empty sets do not contribute any elements to unions, we can assume that none of the sets in our given countable collection of countable sets is an empty set. If there are no sets in the collection, then the union is empty and therefore countable, Otherwise let the countable sets be As,...
1. Show that if A and B are countable sets, then AUB is countable. 2. Show...
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us
5- Recall that a set KCR is said to be compact if every open cover for...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
JUST DO QUESTION 4 Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École...
JUST DO QUESTION 4
Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École de science informatique et de génle électrique uOttawa School of Electrical Engineering and Computer Science Canada's universiry ELG 3126 RANDOM SIGNALS AND SYSTEMS Winter 2018 ASSIGNMENT 1 Set Theory (due at 11.30 AM Thusday, Jan. 18 in class) I. Your University of Ottaa stdent number has k distinct digits in it. State the set of t and all the subsets of this set that...
Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can...
Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can be written as a disjoint union in the form A ∪ B = (A \ (A ∩ B)) ∪˙ (B \ (A ∩ B)) ∪˙ (A ∩ B). Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle. Exercise 1.10. Prove for...
all parts A-E please. Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some numb...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
Question 7 Classify each of the following sets as finite, countable infinite, or uncountable (no proof...
Question 7 Classify each of the following sets as finite, countable infinite, or uncountable (no proof is necessary): A=0 B = {2 ER: 0 < x < 0.0001} C=0 D=N E = {R} F= {n EN:n <9000} G=Z/5Z H = P(N) I= {n €Z:n > 50 J=Z Bonus: Give an example of a set with larger cardinality then any of the above sets.
b and c please explian thx i post the question from the book Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Def...
b and c please explian thx
i
post the question from the book
Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E e Fo the set function P by ¡f E is finite, 0, if E is finite 1, if Ec is finite. P(h-10, (b) If is countably infinite, show P is finitely additive but not-additive. (c)...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
Identify the correct steps involved in proving that the union of a countable number of countable sets is countable. (Check all that apply.) Check All That Apply Since empty sets do not contribute any elements to unions, we can assume that none of the sets in our given countable collection of countable sets is an empty set. If there are no sets in the collection, then the union is empty and therefore countable, Otherwise let the countable sets be As,...
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
JUST DO QUESTION 4
Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École de science informatique et de génle électrique uOttawa School of Electrical Engineering and Computer Science Canada's universiry ELG 3126 RANDOM SIGNALS AND SYSTEMS Winter 2018 ASSIGNMENT 1 Set Theory (due at 11.30 AM Thusday, Jan. 18 in class) I. Your University of Ottaa stdent number has k distinct digits in it. State the set of t and all the subsets of this set that...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
Question 7 Classify each of the following sets as finite, countable infinite, or uncountable (no proof is necessary): A=0 B = {2 ER: 0 < x < 0.0001} C=0 D=N E = {R} F= {n EN:n <9000} G=Z/5Z H = P(N) I= {n €Z:n > 50 J=Z Bonus: Give an example of a set with larger cardinality then any of the above sets.
b and c please explian thx
i
post the question from the book
Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E e Fo the set function P by ¡f E is finite, 0, if E is finite 1, if Ec is finite. P(h-10, (b) If is countably infinite, show P is finitely additive but not-additive. (c)...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
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