JUST DO QUESTION 4 Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École...
answer question 5 please 3 and 4 are just included to refer to the theorems 3 Prove the following theorem: Theorem 2.2. Let S be a ser. The following statements are equivalent: (1) S is a countable set, i. e. there exists an injective function :S (2) Either S is the empty ser 6 or there exists a surjective function g: N (3) Either S is a finite set or there exists a bijective function h: N S (4) Prove...
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 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...
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...
please explain it step by step( not use the example with number) thanks 1. Determine whether each of these sets is countable or uncountable. For those that are countably infinite, prove that the set is countably infinite. (a) integers not divisible by 3. (b) integers divisible by 5 but not 7 c: i.he mal ilullilbers with1 € lex"Juual reprtrainiatious" Du:"INǐ lli!", of all is. d) the real numbers with decimal representations of all 1s or 9s. 1. Determine whether each...
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...
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
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,...
cept of a randon PROBLEMS 1.1-1. Specify the following sets by the rule method. A= (1,2,3), B = (8, 10, 12. 14), C (1, 3, 5, 7,... 1.1-2. Use the tabular method to specify a class of sets for the sets of Problem 1.1-1. uncountable, or finite or infinite. A (1), B= (x= 1}, C ={0 < integers), D = (children in public school No. 5), E={girls in public school No. 5), F = {girls in class in public 1.1-3....