For an index set I and indexed families of sets {Ai | i ∈ I } and {Bi | i ∈ I }. For each of the following statements, either prove it or find a counterexample. ∪i∈I (Ai ∩ Bi ) ⊆ (∪i∈I Ai ) ∩ (∪i∈I Bi )

Q: Let A, B be set with A non-empty. (a) Prove that if A C B, then A, B are not disjoint. (b) What if…

A:

Q: Theorem 2.8: The closed subsets of R have the following properties: (a) R and Ø are closed sets. (b)…

A: Note : A set is said to be closed if it's complement is open. Theorem used: Arbitrary union of open…

Q: Argue that a set A is a Gδ set if and only if its complement isan Fσ set.

A: The given statement is of Baire’s Theorem.

Q: Suppose A and B are sets. Au B = B if and only if As B. True False

A: Given statement is A and B are two sets. A∪B=B if and only if A⊆B. We know that if A⊆B and B⊆A then…

Q: Let A. B and C be sets. In each part, either prove the statement or give a counterexample. (a) AUBCC…

A: Given: A, B, and C are sets. We have to either prove the statement or give a counterexample. (a)…

Q: Proposition 1.43. A is the intersection of all closed subsets of X which con- tains A. Proof.…

A:

Q: Let A, B, and C be sets (a) Prove that ACB iff A – B = Ø. (b) Prove that if A C BUCand ANB = Ø, then…

A:

Q: Use an element argument to prove the following statement.. Proof: For all sets A, B, and C, An (B-C)…

A:

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: Let A and B be sets. Prove that A c B if and only if A NB = A.

A:

Q: Prove that if A is transitive set, then A+ is also a transitive set.

A:

Q: Suppose that X is a set and {A. : a E I} is an indexed family of sets (which contains at least one…

A:

Q: Prove that for all sets A and B, A – B = AN B

A:

Q: (b) Let = {A; : i EN} be an indexed family of sets with the property that A₁ A₂ 22 A, 2..., that is,…

A:

Q: Let A and B be subsets of some universal set U. Prove each of the following: *(a) An BCA * (b) A…

A:

Q: (3) Use an element proof to prove that for all sets A, B, C, and D, if An C =0 then (A x B)n (C x D)…

A: The proof of the given statement is given in the explanation part.Explanation:Steps and explanations…

Q: F oranges pineapples bananas apples pears V peas carrots beans corn potatoes

A:

Q: Jive R² which are example of two subsets homeomorphic. not C specify the set explicitly) g

A:

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: Assume all sets are subsets of a universal set U. Suppose that A, B, and C are arbitrary sets, Prove…

A: Given, A-C∪B-C=A-Bc∪C

Q: Prove that the following 2 sets are equal: {4k + 5 : k ∈ Z} = {4k + 1 : k ∈ Z}.

A: Two sets are equal if and only if They have the same element  We are using here if x €A. <=>…

Q: Find sets A, B and C such that A ∩ B,    B ∩ C and A ∩ C are non-emptysets and A ∩ B ∩ C = φ.

A:

Q: If A and B are sets such that ACB then A × BC B x B. true false

A:

Q: Let A, B, and C be sets. Use the the identity A − B = A ∩ B, which holds for any sets A and B to…

A: Identity Used: A-B=A∩BC Property Used: Intersection of sets is associative A∩(B∩C)=(A∩B)∩C

Q: Let U= {1,2,. 1} be the universal set. Let A={x|x eU,x is a multiple of 6, B={x\xeU,xis a multiple…

A: Given: U={1,2,...16} A={x|x∈U, x is a multiple of 6} B={x|x∈U, x is a multiple of 5}…

Q: 2. Let A and B be finite sets. Prove that A B if and only if |A| = |B|. %3D

A:

Q: If A and C are nonempty classes, prove that AC B and C C D if and only if AxC C BxD.

A:

Q: V Let A. B be sets Prove that AnB = if and only if A B

A: