Give an example to show arbitrary union of open sets is again open. Is the result true for arbitrary intersection?

Q: i. Provide an example to prove that the arbitrary intersection of open sets is not an open set ii.…

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Q: Define Open set ?

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Q: 3.1 Show that a finite intersection of closed sets is closed.

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Q: True or False The intersection of two sets is always a subsetof their union

A: To check the statement

Q: Using contradiction method, prove that for every nonempty sets A and B, An (B- A)=0.

A: The given problem is to prove the following set identity using the contradiction method.

Q: 26

A: The objective is to find an infinite collection of closed sets whose union is not closed.

Q: The boundary of a set is equal to the boundary of the complement True O False O

A: A point of a set is called boundary points if any neighbourhood of this point contains point of the…

Q: can two disjoint sets be close? Justify your answer using an example.

A: We know that any closed interval of the form [a,b] is an closed set. We use this fact further.

Q: A closed set in the finite complement topology on R.

A: Def: A set C is closed in topological space, if complement of C is open set.

Q: Can sets have different names? Give a precise definition of a set. Give two different notations…

A: A set can be defined in mathematical terms or not in mathematical terms. A way of representing sets…

Q: Explain and give a simple example that shows that the intersection operation on sets is symmetrical.

A: Answer

Q: Give example open set is compact

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

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Q: What is meant by the Cardinality of a finite set? Give one example.

A: The cardinality of a set is meant by the number of elements of the given set. If the set has no…

Q: what is a connected dominating set ? And give examples

A: A connected dominating set of a graph G is a set D of vertices with two properties: 1) Any vertex in…

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: Arbitrary unions of closed sets are closed.

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Q: Using the fact that "a countable union of countable sets is countable", show that the statement that…

A: Please see the below picture for detailed solution.

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

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Q: Prove using the properties of set operations: a. ((AUB)–C)° = (A–C) – (B – C)

A: Solution

Q: Show that the set given in the image below is non convex. Please prove the result mathematically…

A: A set is said to be convex if for any two points in the set the line segment joining those points is…

Q: b. Define the finite and infinite sets in your own words and give example for each set.

A: We use the definition of finite and infinite set to answer the given question.

Q: 3. Using the definition of open sets, prove that an interval (0, a) is open where a > 0 is an…

A: (3) Definition of open sets: To define open set first we define an open ball For a real number x and…

Q: Prove that every Union of Open sets is also Open.

A: To prove: Every Union of Open sets is also Open.

Q: Determine if the union/intersection of two open sets is always open then determine if the…

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Q: What is meant by Union of sets?

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Q: Give an example of a countable collection of bounded sets whose union bounded, and an example where…

A: Let A1 and A2  are bounded sets

Q: Assume that the statement applies to all sets. Is it true or false? A – (BO C) = (A – B) (A – C) 2.

A: A-(B INTERSECTION) C= (A-B)INTERSECTION(A-C)

Q: Give an example of a set of numbers that is: a. Closed under both addition and multiplication. b.…

A: a. Set of real numbers b. Set of odd natural  numbers c. Set of irrational numbers

Q: Proving a divisibility relation is transitive

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Q: The empty set is open set

A: See the detailed solution below.

Q: Fill in the blank so that the resulting statement is true. The number of distinct proper subsets of…

A: To fill in the blankThe number of distinct proper subsets of a set with n elements is ____

Q: Provide examples of sets A and B where strict inequality sup(A+B) < sup A+ sup B holds.

A: We have to given an example to prove given inequality:

Q: (a) The intersection of an arbitrary collection of closed sets is closed. (b) The union of any…

A: (a) We have to prove that the intersection of an arbitrary collection of closed sets is closed. Let…

Q: Prove directly that the union of two disjoint sets, where one is countably infinite and the other is…

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Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set,

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Q: Please prove the statement below: For all sets X, Y, and Z subsets of a universal set U (X -- Y) ∩…

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Q: (a) Set-builder form: X X is a natural number andX25 Roster form:

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Q: ) Find an example of an infinite collection of closed sets whose union is not closed.

A: Any closed interval is closed in R. From that we created this example.

Q: Give an example of a set of numbers that is: a.Closed under both addition and multiplication.…

A: (a) Set X is closed under addition if any two elements x, y in X such that x + y in X  Set X is…

Q: 4.3 Give an example to show that the arbitrary intersection of open sets is not an open set.

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

Q: Which of the following sets are closed under multiplication?

A: Given: (a) The set of all prime numbers. (b) The set of all even numbers. (c) The set of all…

Q: A set X and a topology on X with a subset that is both open and closed.

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Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: The union of the universal set and any of its subset is * none of the choices cannot be determined…

A: our objective is to conclude

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

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