Give an example to show that a union of countable sets need not
be countable. (Obviously your example must involve infinitely many
sets.)
Give an example to show that a union of countable sets need not be countable. (Obviously...
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
2. Prove that a finite union of compact sets is compact. Give an example of a countable union of compact sets which is not compact. Book Problems: Chapter 2, Problems 12, 13, 16, 17, 19, 22
Using Baire Category Theorem to show
A Gδ set is the countable intersection of open sets. An Fσ sets
is the countable union of closed sets.
Fo # Gs, and GS UFO # Gso n Fos.
4. Do each of the following: (a) Show that a finite union of compact sets is compact, i.e. given compact sets K1,.., Kn show that K1U .U Kn is compact. (b) Show that an arbitrary intersection of compact sets is compact, i.e. given compact sets {Ka}a where each Ka is compact, show that no Ka is compact. 1 Give a counterexample for (a) in the case that the word finite is replaced by the word infinite, i.e. exhibit infinitely many...
O wor Question 2 5 RQ True False Question 3 5p The union of a countable family of countable sets can be uncountable. True False 5 pt: Question 4
O wor Question 2 5 RQ True False Question 3 5p The union of a countable family of countable sets can be uncountable. True False 5 pt: Question 4
Give an efficient algorithm to compute the union of sets A and B, where n = max(|A|, |B|). The output should be an array of distinct elements that form the union of the sets, such that they appear exactly once in the union. Assume that A and B are unsorted. Give an O(n log n) time algorithm for the problem.
Give an example of infinitely many sets of real numbers, called
such that all four conditions are satisfied at once. They are: i)
each set is bounded above. ii)
for all m and all n. iii) the intersection of and is empty whenever
m and n are not equal. iiii) for all n, is not an
element of .
Not sure what to do here, but I believe it can be done using the
fact that there is infinitely many...
Give examples of two sets that are not Jordan measurable such that the union is Jordan measurable
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.