Which of the following is true? O Finite intersections of compact sets are compact, but there exists an infinite collection of compact sets whose intersection is not compact. O There exists a finite collection of compact sets whose intersection is not compact. O Arbitrary intersections of compact sets are compact, but there exists a finite collection of compact sets whose union is not compact. O Arbitrary intersections of compact sets are compact, and finite unions of compact sets are comapct, but there exists an infinite collection of compact sets whose union is not compact.

Q: 2. State the definition of a countable set we mentioned in the lecture and prove the following…

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Q: 6) Define the cardinality definition of addition.

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Q: 6. Show that in a topological space, the union of finitely many compact subsets is com- pact. Is the…

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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: Do A = {1,2,3, 4,...} and B = {-1,0,1,2, 3, 4, ...}have the same cardinality? O No because elements…

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Q: 2. Let A be any nonempty set. (a) Give the definition for: "A is not countable." (b) Is it true or…

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Q: Example 5. If A, B and C are three sets and U is the universal set such that n(U) = 1000, n(A) =…

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Q: 2- Every finite set in any topological space is a- Regular set, b- connected set, c- Hausdroff' set,…

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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: For each of the following collections of sets, define a set An for each n N such that the indexed…

A: (a) From given collection of sets,we can say ,In other words, given collection is nothing but now…

Q: Do infinite uncountable sets have the same cardinality? If yes explain why. If no give a counter…

A: No. Infinite uncountable sets may have different cardinality.

Q: Use the method of intersection of sets to compute the following. (a) GCD(12,16,8) and LCM(12,16,8)…

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

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Q: Prove that the cartesian product of a countable set is countable

A: This is question related to set theory's topic countability.

Q: 3. If A is a set, and |A| represents the cardinality of A, then use the definition of set A has the…

A: To prove that the set [1,2) has the same cardinality as the set [3,∞), we need to establish a…

Q: If a finite set A equals the union of k subsets A1, A2,..., Ak, then the number of elements in A…

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Q: 8

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Q: Suppose A1, A2,... An are sets in some universal set U, and n ≥ 2. Prove that(A1 ∪ A2 ∪··· ∪ An)` =…

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Q: For an indexed collection of sets to have an intersection an element needs to be in every set right?…

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