Let A, B, and C be sets. Show that a) (A ∪ B) ⊆ (A ∪ B ∪ C). b) (A ∩ B ∩ C) ⊆ (A ∩ B). c) (A − B) − C ⊆ A − C. d) (A − C) ∩ (C − B) = ∅. e) (B − A) ∪ (C − A) = (B ∪ C) − A.

Q: Let A, B, C, D be sets. Prove that (a) A × (BN C) = (A × B) N (A × C') (b) (A × B) N (C × D) = (AnC)…

A: Given (a) A×B∩C=A×B∩A×C (b) A×B∩C×D=A∩C×B∩D

Q: 2. Find sets A, B, C, D such that the sets (AN B)U (C\ D), ((An B) U C) \ D, An(BUC) \ D, An(BU(C\…

A:

Q: ) (AO B) © (A^ B) = A UB.

A: Introduction :

Q: 2. Find sets A, B, C, D such that the sets (An B) U (C\ D), ((An B)U C) \ D, An(BU C) \ D, An(BU (C\…

A: Example of such set A,B,C,D are given below

Q: Prove that for sets A and B, if A C B then P(A) C P(B

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: Prove that if A1,A2,…,An and B are sets, then (A1−B)∪(A2−B)∪⋯∪(An−B)=(A1∪ A2 ∪⋯∪ An)−B.

A: To Prove- Prove that if A1, A2, ...., An and B are sets, then A1 - B ∪ A2 - B ∪ .... ∪ An - B = A1 ∪…

Q: Prove that if A and B are sets, then P(A) U P (B) C P(AUB).

A:

Q: Show that for all sets A, B and C, If A CB, then B CA.

A:

Q: Let A, B, and C are three sets, prove that Cx (A – B) = (C x A) – (C × B) C.

A:

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (CC n D) U (C n D) = D where Ce…

A:

Q: (b) Show that the set A = {(x, y) E R² : y > 0} is not complete.

A: Given: The given set ,

Q: Show that if A and B are sets with AS B, then a) A UB-B. b) ANB-A.

A: Let A and B are two sets with A⊆B a) Let x∈A∪B which implies that x∈A or x∈B. Hence we can imply…

Q: 4. In each case, sketch the closure of the set: 1 (c) Re (d) Re(z?) > 0.

A: Given:

Q: What is the intersection of sets A and B?

A: The intersection of sets is a set which include the common of the two sets A and B. It is…

Q: AAB = (A\ B) U (B\ A). Prove or disprove: (1) (2) (AAB) (AAB)° (AUB) \ (ANB) A¢AB° %3|

A:

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: Prove: If = ¿ (where a, b, c, and d are nonzero), b then d*

A: Given:               ab=cd  (where a,b,c  and d are non zero.)

Q: Prove that for any sets A, B, C, ANB = AN C and AUB= AUC if and only if B = C. %3D Prove that for…

A:

Q: Let A, B, and C are three sets, prove that C × (AN B)' = (C X A') U (C × B') CS

A:

Q: Let A, B, and C be sets. Prove A × (BnC) = (A × B)n (A × C).

A:

Q: Disprove the following statement by finding a counterexample. For all sets A, B, and 6, C, An (BUC)…

A:

Q: Suppose that each of A, B, and C is a set and each of BnC, A–(BnC), A-B, A-C, and (A – B) n (A – C)…

A:

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

A: Let A, B and C are sets. We have to find out that A-B = A- C implies B = C or not If not then we…

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (Cºn D) U (C N D) = D where CC…

A: Identities of the sets

Q: Prove the following theorem directly. 'Let B, E,W be sets. If B Ɔ W, then BU E WUE"

A: Given B, E and W are three sets .

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: If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10…

A:

Q: Let A, B, and C are three sets, prove that Bx(AUC)=(B× A)U(B×C)

A: "Since you have posted multiple questions, we will answer the first question. If you want the…

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (C° n D) U (Cn D)= D where Ce…

A:

Q: Let A,B,C,D be sets. Prove that (A∩B)×(C∩D)=(A×C)∩(B×D)

A:

Q: Find the intersection of the sets. {r, e, a, l}∩{l, e, a, r}

A: Given:A=r,e,a,lB= l,e,a,rIntersection set:The set containing all elements of A that also belong to…

Q: 1. Let A, B, and C be sets. Does A - B = A – C imply B = C? If so, prove it. If not, give a…

A: This question is related to Set theory, We will solve it using given information.

Q: If sets X union Y are countable, then prove or disprove that both X and Y are countable.

A:

Q: What is the union of sets A and B?

A: Assume the independent sets A and B. Set A = 0,1,2Set B = 4,5,6

Q: Let A, B, and C are three sets, prove that B×(AnC)=(B× A)^(B×C)

A: Given 3 sets A, B and C. We have to prove that B×A∩C=B×A∩B×C. For this, we have to show that…

Q: Let A and B be two finite sets such that |A| = 20, |B| = 28 and |A U BI

A: Given A=20B=28A∪B=36

Q: Is the following set connected: V = {(x, y) € R² | 1<x² + y² <4}? Briefly justify your answer.

A: Connected set:- If any two points of a given set can be joined by a continuous polygonal arc such…

Q: Are the sets Z x Z; ((x.y): x*y?) closed in (R , | |) ? What about the sets (0,1]U(2,3); [0, 00) in

A:

Q: Let A, B, C be three sets. Show that if B C A and C C A \ B, then BC A \C.

A: Given: Let A, B, C be three sets. To show: If B⊆A and C⊆A \ B, then B⊆A \ C.

Q: Give an example of two sets A and B which are homeomorphic, A is complete and B is not.

A:

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

A: Here we will prove or disprove the given statement.

Q: we defined the symmetric difference of sets A and B to be A AB = (A U B) – (A n B) = (A – B) U (B –…

A: The proof of associative law for symmetric difference is given as follows

Q: 3. Consider three finite sets A, B and C. Show that: (a) (b) (c) (AB) x C = (A ×C) \ (B × C ); (AUB)…

A:

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

A:

Q: What is the size of the symmetric difference of two finite sets A and B, that is AA BI ? o IIAI -…

A:

Q: In drawing 3 CD’s from a lot, defective and non defective CD’s , let A, B , C and D mean at least…

A: let A, B , C and D mean at least one Defective CD, at least one non defective CD, 2 defective CD’s…

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

A:

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”.