One of the following statements is True a) Z is connected set if there exist two non-empty separated sets A, B such that Z = A U B b) Z is connected set if there exist two disjoint sets A, B such that Z = A UB c) Z is disconnected set if there exist two non-empty separated sets A, B such that Z = A U B d) Z is disconnected set if there exist two disjoint sets A, B such that Z = A U B

Q: Complete the proof: If A≤B and B≤C, then ACCn(transitive property for set in inclusion) Need to…

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Q: • Show that if D is a set and E is a closed set containing D, then E must contain ƏD, i.e., the…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

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…

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Q: In exercise 1 to 3, Consider the following sets. A= {a, b, c} B= {b, c, d, e} C= {b, c, d} U= {a, b,…

A: Union of any two set contains all the elements from both the sets without repetition. Intersection…

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: Answer the following questions by replying to this post. Given that the elements of the sets A, B,…

A: As per the company guidelines we can solve first three parts of question. I hope that the given…

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

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Q: Use the tabular method to to prove the following in general: 1. (An B) C (AU B) 2. The O is a subset…

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Q: Given a universal set U and sets A, B CU, prove that A C Bº if and only if AnB = Ø. Prove that (AN…

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Q: Consider S to be the universal set. Let X = (AUB) nC and Y = (AnT)U (An B). %3D Write out the…

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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…

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Q: Can a set and it's Complement be both nowhere dense in R? A Ac=R-A

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Q: 2. Given A, B, and C are non-collinear points, draw the following or explain why it is impossible…

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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: Find the Cartesian Product of the given sets:  With A = {c, d, f} and B = { w, x}, find B x A.

A: A = {c, d, f} and B = {w, x}, B x A = { (w,c), (w,d), (w,f), (x,c), (x,d), (x,f) }

Q: Let X = {a, b, c, d,e} and let T = {X, ø, {a}, {a,d}, {a,e}}, then * %3D (X, T) is not Hausdorff but…

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Q: Given a universal set U and sets A, B CU, prove that A C Bº if and only if AnB = 0. Prove that (AN…

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Q: If A C C and B C D then A U BC CU D.

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Q: Q: In usual topological space (R.Tu). answer the following: 1- Is [0,1] is compact set in R? Why?…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

Q: Consider the set S = {zeC:0< Re(z) < 2 & 0<Im(z) < 3} What is its interior? {zeC:0<Re(z) < 2 & 0<…

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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: Decide whether the following statements are true or false. If false, give a counterexample. That is,…

A: a). Given, If A⊆B, then nA≤nB. Since A is a subset of B, therefore all the elements of A are also…

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…

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Q: suppose R has of the set inclusions: (a) Ba(1) C Ba(0) in R (b) B2(1, 1) C Ba(1,0) in R? (c)…

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Q: (a) prepare the list of the eler separate Venn- diagrams. ii. (PUQ)NR Ing sets by drawing iii. PUQ…

A: 4. (a) (i) Venn diagram of P∪Q is,

Q: • Show that if D is a set and E is a closed set containing D, then E must contain ðD, i.e., the…

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Q: .et A, B be any two sets. Prove that if A C B, then A n B = p.

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Q: Prove the following statements about sets A and B. (AUB)° = A° Bº

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Q: Let X = (6,7), and d be the ususal mortric, d(r, y) = |r - yl Gire ezample of a set ACX, to be not…

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Q: Suppose A, B, and C are sets, A. Prove that: Ax (Bn C) = (A × B) n (A × C) %3D Using Membership…

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Q: Suppose that (x, y) E (A \ B) × C, for sets A, B,C C S. Then a E A and r¢ B, and y E C. This implies…

A: Use the following definitions, to prove the given result. Let A and B be two sets. The Cartesian…

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Set C and the universal set U are defined as follows. U={p,q,r,x,y,z} C={p,q,x}

A: Given,        Set C and the iniversal set U are defined as follows :         U=p,q,r,x,y,z…

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Let A and B be sets. Prove or disprove that B – A = BU (AnB)

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Q: 1. Prove the following statements about sets A and B. A B= AN Bº %3D

A: Given: Let A and B be two sets. To prove,A-B=A∩Bc

Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

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Q: Let Cc R be a connected set and a , be R. Prove that [a , b] c C.

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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: Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1)are in A × B, find…

A: In AxB the first element will be from A and the second element will be from B So, x,y,z are in A And…

Q: d) Let X, Y, Z be sets. Calculate |X UYU Z| if |X| = 50, |Y| = 90, |Z| = 71, |X n Y| = 8, |Xn Z| =…

A: Given that X, Y and Z be the three sets.

Q: Prove by contradiction that for any sets A and B , if A∩(B∪C)= ∅ and B⊆C, then B-(A∪C)=∅.

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Q: Is it true that (A×A)∖(B×B)=(A∖B)×(A∖B) for any two sets A, B? If it is true, prove it. Otherwise,…

A: Explanation of the answer is as follows

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

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Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…