Prove that the union of two disjoint sets, where one is countably infinite and the other is finite, is countable.

Q: Suppose that S is a non-empty finite set. Prove that ? has as many even-sized subsets as it does…

A: Let S be a non-empty set. Let A be the set of even sized subsets of S and let B be the set of odd…

Q: Show that compact sets are closed under arbitrary intersections and finite unions. (Here we're…

A: To show that compact sets are closed under arbitrary intersections and finite unions

Q: Give two different but equivalent definitions for a set CCR" to be closed, and prove they are…

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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: Show that any finite union of Nowhere Dense Sets is Nowhere Dense.

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Q: Here is a conjecture: Card (A - B) = Card(A) - Card (B) Decide whether this conjecture is true or…

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Q: prove that Cantor set is closed and compact.

A: Use the property that arbitrary intersection of closed set is closed and the Heine Borel Theorem.

Q: Set is closed

A: To prove that the Cantor Set is closed.

Q: Is the Cantor set closed? Is it compact? Explain!

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Q: Let A and B be two sets. a) Suppose that A and B are both countably infinite sets. Prove that there…

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Q: |(A ∪ B)\(C ∪ D)|

A: We have to give a formula for |(A ∪ B)\(C ∪ D)| in terms of the sizes of the four sets A, B, C, D…

Q: Prove T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set

A: We have to prove the following -   T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set.

Q: Can a topology on an infinite set not be discrete and indiscrete in which every subset is clopen?

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Q: Are the following true or false? If true, prove. If false, give a counterexample. (a) The…

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Q: Give an example of sets {An}n≥1 such that for each n ≥ 1, the set An ⊂ [0, 1] is both compact and…

A: Construct sets Ann≥1 such that for each n≥1, the set An⊂0, 1 is both compact and connected, but the…

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

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: open

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Q: Prove that a finite union and an arbitrary intersection of compact sets is a compact set.

A: Step 1: Step 2: Step 3: Step 4:

Q: Consider two sets A and B defined in the universal set U. Set A contains elements that are prime…

A: Prime numbers less than 20 (Set A): A prime number is a natural number greater than 1 that has no…