Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.

Q: Prove that every set D ⊆ ℕ that is definable in N := (ℕ, 0, S , +, ·) is actually ∅-definable.

A: A set D ⊆ ℕ that is definable in N := (ℕ, 0, S , +, ·) refers to a set that can be described or…

Q: Let A, B, and C be subsets of a universal set U and suppose n(U) = 150, n(A) = 27, n(B) = 29, n(C) =…

A: We have given  Let A, B, and C be subsets of a universal set U   n(U) = 150, n(A) = 27, n(B) = 29,…

Q: 6) Define the cardinality definition of addition.

A:

Q: Can a set and its complement be both dense in R? Can a set and its complement be both nowhere dense…

A:

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: Let X={a,b,c} and Y=∅. What is X∩Y? If the set is empty type ∅, otherwise write your answer in…

A:

Q: The family of all finite subsets of a countable sets is countable.

A: Countability

Q: 5. Give an example of a collection (A,, of sets with the properties that: (a) A, # Ø for each i> 1,…

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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: 5. Show that a finite union of countable sets is countable.

A: The answer is given as follows :

Q: Give an example of 4 non-empty sets, A, B, C and D such that all the following are true: A ∩ B ̸ =…

A: We have to give an example of 4 non-empty sets A, B, C, D such that all the following are…

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…