Let A be an uncountable set and let B be the set of real numbers that divides A into two uncountable sets; that is, s ∈ B if both {x :x ∈ A and x < s} and {x : x ∈ A and x > s} are uncountable. Show B is nonempty and open.

Q: Prove for all sets of A, B, that P(A) U P(B) is a subset of P(AUB).

A:

Q: Let A be a set and let B be a proper subset of A. Is it possible for B to have the same cardinality…

A: Given: Let A be a set.And B be a proper subset of A.

Q: Prove that for any sets A, B, C: (AUB)-B A - B

A:

Q: D. Let A, B, and C be subsets of some universal set U (a) If ANBC C and A° N B C C, prove that B CC

A:

Q: Let A and B be any sets (subsets of some universal set U). Use a proof by contradiction (and…

A: The given problem is to prove the following statement by contradiction method. In this method , we…

Q: Let A and B be subsets of a universal set U and suppose n(U) = 300, n(A) = 150, n(B) = 120, and n(A…

A: Please see the below explanation.

Q: Let A and B be subsets of a set X. Prove that if A ⊆ Bc, then A and B are disjoint.

A:

Q: Let A, B sets such that A ¢ B and B C C. Prove that A ¢ C.

A:

Q: Does there exist a non-empty open set A and two elements in A that are farthest apart (among all…

A: There are no two elements that are farthest apart in a non-empty open set. Justification : Let A…

Q: Let A and B be sets and suppose that A U B = B. Prove that ACB.

A:

Q: Find a counterexample to show that the statement is false. Assume all sets are subsets of a…

A:

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

A:

Q: Let A {a1, a2, ...} and B = {b1,b2, ...} be countably infinite sets. %3D Prove that A × B is a…

A:

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

A:

Q: Let A and B be sets. Show that A∪(A∩B) =A using a traditional verbal proof, a series of equalities…

A:

Q: Let B and C be the subsets of some universal set X then prove that (C N B')' = BU C'

A:

Q: Find a counterexample to show that the statement is false. Assume all sets are subsets of a…

A: The solution is as shown below

Q: For an element to be in the intersection of sets A and B, the element must be in set Am in set B.…

A: Any elements of A intersection B must be in set A and in set B

Q: 2. Prove that if A, B,C are sets such that ANBNC = 0, then (A\B)n(C\B) = AnC. Prove that for onu got

A: Recall that, set A\B means a set of elements of A that does not belong to B.

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

A:

Q: Let A, B be non-empty bounded sets, and A Ç B (A is a proper subset of B). Then sup A < sup B. True…

A: Let A=(1,2] and B=[1,2]

Q: If sets X and Y are countable, then prove or disprove that X-Y is uncountable.

A:

Q: 4. Let A and B be sets. (a) If A and B are subsets of the same universal set U (so A × B is a subset…

A: The definition of complement is given at the end of proof

Q: Let A and B be nonempty sets of real numbers. If sup A exists then: a. sup (A U B) exists if sup(B)…

A:

Q: Let A and Bbe subsets of a universal set U and suppose n(U) = 400, n(A) = 200, n(B) = 160, and n(A N…

A:

Q: Prove by giving an example that if any set A, g. l. b(-A) = - l. u. b(A

A: Take a set A with two elements 1 and -1

Q: Let A and B be finite sets such that B = {b1, b2}. Prove that |A – B| < |A|.

A: We are given that A and B are two finite sets, where B= {b1, b2}.We have, |B| = 2, and let |A| =…

Q: Prove that A is an infinite set if and only if ∀n ∈ ω, A has a subset B such that B ≈ n.

A: given that A is an infinite set let B be any set let prove this contradiction suppose assume that B…

Q: Let A and Bbe subsets of a universal set U and suppose n(U) = 400, n(A) = 200, n(B) = 160, and n(A N…

A:

Q: uppose that ℱ is a non-empty family of sets, B is a set, and ∀A ∈ ℱ (A ⊆ B). Is ∪ℱ ⊆ B? Either…

A: Given: To explain the given statement as follows,

Q: Let A, B sets such that AnB = An C. Prove that B = C

A: .

Q: Use the definition of set operations to prove that for any set A, A- A = Ø . A + A = Ø

A: To prove: Using definition of set operations, for any set A prove that                  a) A-A=ϕ…

Q: Prove that if A and B are sets, then A - B = An BC.

A:

Q: Let C be a nonempty subset of a bounded set D. Why does inf C and sup C exist? Show that (c) inf D ≤…

A:

Q: If A is a countable set and B is an uncountable set, then the MOST we could ever say about A U Bis…

A: We have given the three MCQ questions about countability and uncountability . We have to choose the…

Q: Consider the following theorem of set theory: For any two sets A, B, [A U (B – A)] = A U B. a.…

A:

Q: Prove that for any integers N and m, the set 1 x + m 1 {x € A : \x| < N, and (x inA = {x}} | m…

A:

Q: Prove that if A is a set then so is {A} but do NOT use an argument that involves stages explicitly.

A: 3 Given A is any set. We have to prove that A is also a set. Since, A is any set Then as every set…

Q: Let Ω be a set and A = {A ⊂ Ω : A is finite or Ac is finite }

A: given that Ω is a set and also given A ⊂Ω if Ω contains n elements  then A contains (n-1) elements…

Q: Let A and B be subsets of a universal set U and suppose n(U) = 300, n(A) = 150, n(B) = 120, and n(A…

A: We have given the following n(U) = 300, n(A) = 150, n(B) = 120, and n(A ∩ B) = 60.

Q: let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are…

A: Given that X=a,b,c. To construct two topologies that have at least four open sets. We know that in…

Q: For every positive integer n, prove that a set with exactly n elementshas exactly 2n subsets…

A:

Q: what is the proper subset of set A given that A= { x ∈ Z+ /2 < x < 6}? note: Z+ - is the set of…

A:

Q: supC = supA × supB.

A: This is a question from real analysis with usual topology.

Q: Let A and B be subsets of a universal set U and suppose n(U) = 400, n(A) = 200, n(B) = 160, and n(A…

A: A set is a collection of elements. The elements that make up a set can be any kind of mathematical…

Q: Show that if A is an infinite set, then whenever B is a set, A ∪ B is also an infinite set.

A:

Q: Give an example of a set A such that there is a set B with B as the element of A but B is a subset…

A: Give an example of a set A such that there is a set B with B as the element of A but B is a subset…

Q: Show that the union of two sets, each of measure zero, has measure zero

A:

Q: Prove that for all sets A & B, (B-A) = A intersects B^c

A: