Homework Answers
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, w...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si
3. (20 pts) Let ụ be a finite set, and let...
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...
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the con...
Let X be a set and let T be the family of subsets U of X such
that X\U (the complement of U) is at most countable, together with
the empty set. a) Prove that T is a topology for X. b) Describe the
convergent sequences in X with respect to this topology. Prove that
if X is uncountable, then there is a subset S of X whose closure
contains points that are not limits of the sequences in S....
Part(c) and partd(d) 1.34 Let A. A.. be a countable collection of subsets of U.Prove the...
Part(c) and partd(d)
1.34 Let A. A.. be a countable collection of subsets of U.Prove the following. a) If B cU, An, then B U,(An n B). b) If Un An-U, then E U(An E) for each subset E of U c) If Ai, A2,... are pairwise disjoint, so are Ain E, A2n E,... for each subset E of U d) We say that Ai, A2,... form a partition of U if they are pairwise disjoint and their union is...
Question 6. a) Let A, B and C be open subsets of R. Prove that AnBn...
Question 6. a) Let A, B and C be open subsets of R. Prove that AnBn C is open. b) Give an example to show that the intersection of infinitely many open sets may not be open.
For the following set X and collection T of open subsets decide if the pair X,...
For the following set X and collection T of open subsets decide if the pair X, T satisfies the axioms of a topological space. If it does, determine whether X is connected. If it is not a topological space then explain which axioms fail. X = Z and a subset U ⊂ Z is open if and only if its complement Z \ U is finite or U = ∅.
Let X be a finite set and F a family of subsets of X such that...
Let X be a finite set and F a family of subsets of X such that every element of X appears in at least one subset in F. We say that a subset C of F is a set cover for X if X =U SEC S (that is, the union of the sets in C is X). The cardinality of a set cover C is the number of elements in C. (Note that an element of C is a...
4. Ranking/Unranking Subsets. Let A be a set of n elements and set Sk(A) be the collection of all k-element subsets of A. Recall that |Sk(A)I - (a.) (8 points) Describe a ranking algorithm to rank a...
4. Ranking/Unranking Subsets. Let A be a set of n elements and set Sk(A) be the collection of all k-element subsets of A. Recall that |Sk(A)I - (a.) (8 points) Describe a ranking algorithm to rank a k-element subset of an n-element set. (b.) (8 points) Describe an unranking algorithm to unrank an integer 0 < s< [into a ithm to unrank an integer 0 S s <C) k-element subset of an n-element set. (c.) (10 points) As examples, let...
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...
Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partiti...
Prove the theorem
Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partition). If the elements of Ω are pairwise disjoint, then ~() is transitive.
Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partition). If the elements of Ω are pairwise disjoint, then ~() is transitive.
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si
3. (20 pts) Let ụ be a finite set, and let...
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...
Let X be a set and let T be the family of subsets U of X such
that X\U (the complement of U) is at most countable, together with
the empty set. a) Prove that T is a topology for X. b) Describe the
convergent sequences in X with respect to this topology. Prove that
if X is uncountable, then there is a subset S of X whose closure
contains points that are not limits of the sequences in S....
Part(c) and partd(d)
1.34 Let A. A.. be a countable collection of subsets of U.Prove the following. a) If B cU, An, then B U,(An n B). b) If Un An-U, then E U(An E) for each subset E of U c) If Ai, A2,... are pairwise disjoint, so are Ain E, A2n E,... for each subset E of U d) We say that Ai, A2,... form a partition of U if they are pairwise disjoint and their union is...
Question 6. a) Let A, B and C be open subsets of R. Prove that AnBn C is open. b) Give an example to show that the intersection of infinitely many open sets may not be open.
4. Ranking/Unranking Subsets. Let A be a set of n elements and set Sk(A) be the collection of all k-element subsets of A. Recall that |Sk(A)I - (a.) (8 points) Describe a ranking algorithm to rank a k-element subset of an n-element set. (b.) (8 points) Describe an unranking algorithm to unrank an integer 0 < s< [into a ithm to unrank an integer 0 S s <C) k-element subset of an n-element set. (c.) (10 points) As examples, let...
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...
Prove the theorem
Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partition). If the elements of Ω are pairwise disjoint, then ~() is transitive.
Theorem 6.59. Let A be a set and let Ω be a collection of subsets of A (not necessarily a partition). If the elements of Ω are pairwise disjoint, then ~() is transitive.
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