Show that for any sets A and B,A = ( A ∩ B ) ∪ ( A – B ) and                    A ∪ ( B – A ) = ( A ∪ B )

Q: Give examples of sets A, B and C such that: (a) A ∈ B and B ∈ C, and A /∈ C (b) A ∈ B and B ∈ C, and…

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

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Q: Let A, B, C be three non-empty sets. Suppose that A× C = B× C. Prove that A = B

A: Here given that A, B, C be three non-empty sets. And suppose that A×C=B×C. We have to prove that…

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

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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.

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

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Q: Suppose that A and B are sets. Prove that A- B = AnB.

A: Suppose that A and B are set. Prove that A-B=A∩B¯.

Q: Prove that for sets  A and B,  we have  (A∪B)−(A∩B)=A−B if and only if B⊂A.

A: By theorem we know that,            A⊂B if and only if A∩B =A and  A∪B =B

Q: The set of elements common to both set A and set B is called the______ of sets A and B, and is…

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Q: If A and B are non-empty sets defined on a universal set U. and An B = Ø, then A\B is equal to

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Q: For any sets A, B, and C, prove that Ac Bnc if and only if A c B and A CC

A: Consider the statement,

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

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Q: The set of elements common to both set A and set B is called the of sets A and B, and is symbolized…

A: The set of elements common to both set A and set B is called the ________ of sets A and B, and is…

Q: Use an element argument to prove that for all sets A and B, (An B) U (AN BC) = A. %3D

A: I have proved that sets on both side are subsets of each other and ten concluded that they are equal

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

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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 formally that A\ B= A\(An B), for any sets A and B.

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

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Q: Use the pick-a-point method to show that if A and B are subsets of some universal set U, then A ∪ (B…

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Q: For any sets A, B,C prove that if BCC then A – (BUC) = A – C. -

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Q: Let C and D are be the subsets of some universal set U. Then D-C = C'nD is

A: We have to identify the statements as true or false.

Q: Prove that if A1, A2, ..., A, are any n > 2 sets, then A NA2 N …… A„ = A¡ UA2 U.…UA..

A: We will  prove the given statement using mathematical induction method.

Q: Let A and B be sets. Prove that the intersection of A and B is is a subset of the union between A…

A: To show that the elements which are common to both A and B belong to either A or B.

Q: Let A, B and C be sets. Use an element argument to prove that (A - B) - C≤A-C.

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Q: Determine whether the following theorem is true or false: For all sets A, B and C, if A is a subset…

A: If X is a strict subset of Y, but X is not equal to Y which means there exists at least one element…

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 if A is a set then so is {A}!

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Q: Prove or disprove: For all sets A,B and C: (A-B)-C = A-(BUC).

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Q: Let A and B be sets. Prove that if A = B, then A+ = B+.

A: Given: A and B To find: Solution

Q: Prove or disprove that for all sets A, B, and C, we have a) A × (B ∪ C) = (A × B) ∪ (A × C). b) A ×…

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Q: Use the element method and contradiction to prove that for all sets A, B, and C, if A C then An…

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Q: For any sets A, B, C  use element argument to prove that  (A-B) - C is a subset of A - C

A: By the element argument we prove the given problem as follows :

Q: The empty set is open set ylgn O İhi O

A: The complement of an empty set is the whole set, which of course contains everything including all…

Q: If A is a countable set and B is a subset of A then

A: If A is a countable set and B is a subset of A then B is also a countable set since, every subset of…

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: Show by membership that for all sets A, B and C A - (AnB) CA-B

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Q: Prove that the following statements are EQUIVALENT: For any sets A and B, a. ACB b. AnB = A C. AUB=B

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Q: Let A,B be sets. Prove that AAB = if and only if A= B

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Q: Let A, B and C be sets. Prove that CCANB if and only if C CA and CC B.

A: Consider the given information: Let A, B and C be sets. To prove that C⊂A∩B if and only if C⊂A and…

Q: 4. Prove that the set Z is a closed set in R.

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Q: Let A, B, and C be sets. Does A – C = B – C imply A = B? If so, prove it. If not, give a…

A: We have to give a counter example for  A – C = B – C imply A = B.

Q: For an element to be in the union of sets A and B, the element must be in set A in set B. but not…

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Q: Prove that if A, B.C are subsets of a universal set U, then AU(BnC) C (AUB)n(AUC)

A: use the basic property of subset and set theory.

Q: Let A and B be sets. Prove that (A − B) ∩ (B − A) = ∅.

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

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.