Let K denote the set of all constant sequences in RN. Prove that relative to the box topology, K is a closed set with an empty interior.

Q: Show for every nonempty, finite set E that supE = maxE.

A: Given, E is a finite set. E is a non-empty set. To prove, supE = maxE We know that, sup is supremum…

Q: In the Axiom Systems section of the Background chapter (1.6), a binary relation <on a nonempty set X…

A: Given X is a set consisting of 64 teams.  Let ⩽ be the relation on X such that for a,b∈X, a≤b…

Q: Exercise 2.16. Suppose that A and B are countably infinite sets. (A) Show that AUB is countably…

A: Suppose A and B are two countably infinite sets.  To prove that A∪B is also countably infinite.…

Q: Find an example of a nonempty set satisfying: 1. Closed, Dense in R 2. Open, Dense in R ~1. & 2.…

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Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: Create a flow proof of each of the following results. Remember in a flow proof each step should be a…

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Q: Prove that (2,5) is an open set

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Q: Prove that if S is a bounded set then S is bounded.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Prove that the set Q \ Z is countable.

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Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

A: Given:   If M and N are both bounded subsets of R. M+N :=m+n : m∈M  and  n∈N

Q: Let R be a relation over a set A = {a,b,c,d}. What kind of order is R?  Note: Select the order that…

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Q: Find an example of a set A with |A| = 3 which contains only other sets and has the following…

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Q: Prove that the unoin of a finite number of bounded sets is bounded.

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Q: Determine whether the following statements are true or false: (a) The empty set ∅ is a subset of {1,…

A: Empty set is a subset of every set. A Power Set is a set of all the subsets of a set.

Q: For any two sets A and B, if m* (A) = 0, then m* (AUB) = m* (B). In particular, if BCA, then m* (B)…

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Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

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Q: = (4) Suppose that A is nonempty set that is bounded from above and let B Prove that B is bounded…

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Q: Suppose A={-1, 0, 4}.Find a set B such that A is equivalent to B.

A: Two sets A and B are said to be equivalent to each other if both the sets A and B have same number…

Q: 2. Find a set A such that A = {2, JA]}.

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Q: Show The union of a finite number of open sets in R is an open set.

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Q: Let A be a partially ordered set that has the least upper bound property. Then every nonempty subset…

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Q: Consider the set A = {xy : x ∈ N, y ∈ N} = {1, 2, 22, 23, 24, ...., 3, 32, 33, 34,......, 4, 42, 43,…

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Q: Let B = {((-1)n*n)/(n+1) : n = 1, 2, 3…}. Find the limit points of B. Is B a closed set? Is B an…

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Q: a and b be disjoint subsets of the universal set U. Assume n(U) = 110, n(A)=50 and n(B) = 25. Find…

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Q: Q.No.2 Let X be a universal set and A, B be the subsets of X.Then prove the following. (a) ((A*…

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Q: Let X be a nonempty set with subsets, A, B, and C. Prove that A ∩ (B \ C) = (A ∩ B) \ (A ∩ B).

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Q: et A be the set of letters in the English alphabet with the usual order. Define (A², <) such that…

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Q: Find all topologies for the set X={a,b,c}. Justify your answer.

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