5. Give an example of a collection (A,, of sets with the properties that: (a) A, # Ø for each i> 1, (b) A, 2 A for each i > 1, and (c) A, = Ø. i+1 i=1

Q: What set is in another class of infinity that is bigger than R?

A:

Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: Can a set and its complement be both dense in R? Can a set and its complement be both nowhere dense…

A:

Q: Suppose that S is a non-empty finite set. Prove that ? has as many even-sized subsets as it does…

A: Let S be a non-empty set. Let A be the set of even sized subsets of S and let B be the set of odd…

Q: 8. Disprove that every compact set has only a finite number of isolated points. V Note that the…

A: Let C be the compact set.If possible suppose that 'C' has a infinitely many isolated points. say…

Q: 2- Every finite set in any topological space is a- Regular set, b- connected set, c- Hausdroff' set,…

A:

Q: What is the class width (i) if one of the class interval is 18-26?

A: If class interval given is 18-26 The class width is difference between upper interval and lower…

Q: 1) Define a subset.

A: To define a subset.

Q: 26

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

Q: 5. Give an example (by words) for a collection of objects which is not well defined

A: Given, We have to write an example for a collection of objects which is not well defined.

Q: If a set A has 63 proper subsets, what is |A|? *

A: Solution...

Q: 2- Prove that H, is convex set.

A: Let us look at an arbitrary closed half-space defined as H+={x∈ℝn:aTx-b≥0}, representing a half…

Q: А - (ВnС) %3D(А — В) n (А — С)

A:

Q: Prove that (2,5) is an open set

A:

Q: What set is in another class of infinity that is bigger than N?

A: We know that N represents set of natural number which is countably infinite. In order to get set…

Q: How many unordered sets are possible that contain three objects chosen from seven?

A: Number of unordered sets that contain three objects chosen from seven = 7C3

Q: 8. Show that the set (0, 1) – {1/2, 1/3, 1/4, -..} is uncountable.

A:

Q: 2- consider the set of all finite subsets of N, is this set countable or not, prove it?

A: Consider the set of all finite subsets of N, is this set countable or not, to prove it.

Q: 4. Let A be an open set in å nörmed space subset B of X .

A: Que  Ans :  Given ( X, ||-||) be a norm linear space,  A be any open subset of X    ( def- set A is…

Q: 8. (a) Show that the intersection of finitely many open sets is open.

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

Q: A universal set U with 117 elements has been partitioned into three subsets, A, B, and C. If B is 2…

A:

Q: If a finite set A equals the union of k subsets A1, A2,..., Ak, then the number of elements in A…

A:

Q: Let A and B be infinite sets that are uncountable. Do an and B have the same car finality as each…

A: Let A and B be infinite sets that are uncountable. Do A and B have the same cardinality as each…

Q: Prove that the unoin of a finite number of bounded sets is bounded.

A:

Q: 5.3.8 Prove that if A CR is a finite set, then A is well-ordered.

A:

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

Q: 15. We have seen that an infinite set can be put into a one-to-one correspondence with a proper…

A:

Q: How many elements does the successor of a set with n elements have?

A: Successor of the set, A is formed by adding an element {A} to the set. So, successor of A will be A…

Q: Calculate the number of subsets and proper subsets for the following set: {x |x is single-digit…

A:

Q: Let {S,},EN is a disjoint seqeuence of measurable sets , then: HUS,

A:

Q: If A is a countable set and B is an uncountable set, then the MOST we could ever say about A U Bis…

A: We have given the three MCQ questions about countability and uncountability . We have to choose the…

Q: 5. Show that a finite union of countable sets is countable.

A: The answer is given as follows :

Q: Consider the set A = {xy : x ∈ N, y ∈ N} = {1, 2, 22, 23, 24, ...., 3, 32, 33, 34,......, 4, 42, 43,…

A:

Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

Q: Give a set that is not closed under addition

A: In the question we have to give a set that is not closed under addition.

Q: Give the inclusion – exclusion formula for four sets where A U B U C U D in terms of sizes of in

A:

Q: 3. Find the number of subsets for set containing 3 elements.

A: Given set  containing 3 element

Q: 6. Give an example of a set S which is (i) bounded above but not bounded below, (ii) bounded below…

A: (i). Let  S = { ..., -5, - 4, - 3, - 2, - 1 }

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

A:

Q: 15 Assume that set S has 11 elements. How many subsets of S have at most 2 elements?

A: Solution : A set S has 11 elements We want to find subset of S have at most 2 elements

Q: ind the all possible number of different multi sets of size '4' from -element set, in which at least…

A: In this question, the concept of permutation is applied. Permutation Permutations are frequently…

Q: prove that the union of a countable set and an uncountable set is uncountable

A:

Q: What is the minimum number of students required in a discrete structures class to be sure that at…

A: As we have five grades: k=5 And the number of students: r=9 And we have to calculate: Nk=r N5=9

Q: 2. (i) Show that given finitely many countable sets A1, A2,, An, the set A1 x A2 x .. x An is also…

A: A set A is said to be countable if its cardinality is same as the set of natural numbers ℕ. If A and…

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: 2- Moke the definitions of open and closed sets. x =Larbic,d,eļ and ノ closed subsets of (x,T) space,…

A: