Lecture 8 Review Exercise: 1) What does it mean if we say that a set S is closed under a binary operation O? 2) What does it mean when we say that a closure property

Q: The set having elements of the complement of the set A will be,

A: We need to find out complement of A.

Q: Show that the set S = {(1, 2, 3), (0, 1, 2), (−2, 0, 1)} spans R3.

A:

Q: Exercise 7.3.2. Let G be the set of subsets of the set {a,b, c}. (a) Does the set G with the…

A: We are authorized to answer three subparts at a time, since you have not mentioned which part you…

Q: ) Now, let's find the regions that represent these sets in general. a)Using the following general…

A: So in this we are given Venn diagram so let's find out required answer.

Q: Give an example to show arbitrary union of open sets is again open. Is the result true for arbitrary…

A:

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

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

A:

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

Q: True or false? For all sets A and B,  (A∪B)'=A'∪B' This can be determined by constructing a…

A:

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

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: C) = (*,') + "- ), Give an interpretation involving subsets. Problem 4) Prove that

A: See the hand written

Q: Let N = {m,y}}. List all the subsets of N. Let A = {j,g}. Is empty set a subset of A? Consider these…

A: Consider the given set  N=m , y Now, the subsets of the set N are,  , m , y , m , y Hence, there are…

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: The subject is on Elements of Set Theory:I really need help and explanation.

A: The symmetric difference of two sets are defined here as,

Q: . Let A and B be two connected subset

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: E is closed. - E is complete, i.

A: It is given that X is a complete metric space and E is a subset of X. We have to prove that the…

Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

Q: Every set S has an open cover.

A: If S is a subset of  R,   Then every set S  will always have an open cover.

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Let A=(0, π ) υ (π e?] %3D c. Is A a closed set? d. Does A contain any isolated points? е. Find the…

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Q: 2. The subset [a,b) of R is neither open nor closed. Prove it and construc an illustration.

A:

Q: Suppose A, B, C are sets. what do you need to show to prove the fllwing definition of subset:…

A: If x is a general element in P ,the given statement implies that each element in P is also in Q…

Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

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Q: Exercise 1. Prove that for sets A, B,C we have C \(AUB) = (C\ A) N (C \ B).

A: We have to show that for sets A, B, C   C ∖(A∪B)=(C∖A)∩(C∖B).

Q: Decide whether the following propositions are true or false.If the claim is valid, supply a short…

A: A compact set is always closed and bounded.

Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

A:

Q: 8. Define two sets, call them A and B, that makes the following statement true; if it is impossible…

A: Define A={1,2} and B={1}.

Q: In Exercises 5 O through 10D, decide whether the object described is indeed a set (is well defined).…

A: (5). The given object is:  n∈ℤ+| n is a large number

Q: (BUA)'= Give a 3 element subset of set B.

A: Given : U = 1,2,3,4,5,6,7,8,a,b,cA = 2,4,6,8B = 1,2,3,4,a,bC = 4,8,a,c

Q: Use membership table to prove (AunionB) – C = (A – C) union (B - C).

A: We need to prove the following statement using membership table. A∪B-C=A-C∪B-C

Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

A:

Q: A set consisting of single point has measure zero .A O nonzero measure .B O open .C O none of…

A: We have to find the measure

Q: Đ A is a closed set.

A: We prove by using the result that intersection of two closed set is closed.

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: Draw a Venn diagram for the symmetric difference of the sets A and B.

A:

Q: 6. Define two sets, call them A and B, that makes the following statement true; if it is impossible…

A:

Q: Let be 7= {x, 0, {a}, {cd}, {a,c,d}; {b, c, d, e}} topology on X = {9₁b, c, d, e}. Check if X is…

A: A set X is connected if and only if X can not be written as union of disjoint open sets. An element…

Q: Write a paragraph that explains why the following proposition is false (Hint: draw a picture and use…

A: Given that A, B, and C are three non colinear points. So the points A, B, and C not lie on the same…

Q: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

A:

Q: Exercise 1.5.5. (a) Why is A- A for every set A? (b) Given sets A and B, explain why A~ B is…

A: (a) To prove that, A ~ A for every set A.

Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: Give example for following 1/ set is not compact

A: Compact set:- A set S⊆R is called compact “if every sequence in S has a subsequence that converges…

Q: (b) Prove that the intersection of finite collection of open sets in the metric space (M, d) is open…

A:

Q: 3A, B E {sets} such that A CBAA= B.

A: To check whether the given statement true or false: ∃A,B∈sets such that A⊆B∧A=B

Q: A collection of open sets in R with the standard topology whose intersection is not open.

A:

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

A: