Exercise 12. Let S ⊆ N be a nonempty subset. Suppose that S has an upper bound B ∈ N. Show that S is finite.

Q: Prove that a subset of a set of measure zero has measure zero.

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Q: Give example of clsad subset of IR cwhise two an dimeasiunal Lebesque outer masuie ís o

A: Consider the unit square L=0,1×0,1. Clearly L is a closed set with boundaries l1, l2, l3 and l4.…

Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

Q: 4. Give an example of a non-measurable subset of R. (witheutProof)

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Q: Define the partially ordered set (N",<) as follows: x < y if Xi < Yi for all 1<i<n. For example, (2,…

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Q: Exercise 2.15. Construct uncountable sets A and B such that A\ B is: (A) finite (B) countably…

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Q: e) Prove the theorem that states that “S" is a countable set, and there exist a surjection of N onto…

A: Two sets which are one-one and onto, have same cardinality. Let S is a countable set,  S=a1,…

Q: Consider all subsets of size four of the set (1,2,3...20) how many of these have a nonempty…

A: Solution :-

Q: 7. Show that if S x T is a closed subset of R?, then S and T are closed subsets of R.

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

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Q: 1) Define a subset.

A: To define a subset.

Q: Suppose (S;)jEN is a sequence of subsets of R", such that each S; C R" has ieN zero content (i.e.,…

A: Option 1, 3, 4,5,6 are true.

Q: Show that the set (1) isopen u k = 1 and find its measure.

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Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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Q: Every infinite subset of R has a cluster point in R. Select one: O True O False

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Q: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: Let M and N be nonempty bounded subsets of R and let M + N := {m + n : m ∈ M and n ∈ N} . Prove that…

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Q: (3) S° is the largest open set in M that contained in S.

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Q: 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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Q: A set SCR" is bounded if there exists k €R such that, for all x E S, || = VE, 0. What can you say…

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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: Exercise 6.40. Let E be a nonempty subset of R that is bounded above. Prove that if a = sup E, then…

A: This is the question of topology.

Q: rcise: Prove that every finite set S of R" has no accumulation point.

A: Proof of the the given theorem is given as

Q: Exercise 1. Let S = [0, 1] - ne N.Then set of limit points of the set S is (a) finite (b) countably…

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E and B is an upper…

A: Let X be a non empty set. Let a∈X. Then a is said to be a lower bound of the set X if a≤x for all…

Q: 3. Let M and N be nonempty set such that M = N, then |M| = |N|. %3D

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Q: shows that there is no topology in X based on the family B = {{a, b}. {a, b, d}, {b, d, c}}

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Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

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Q: compact, then E must be infinite. Exercise 6.46. Prove that if E is a closed subset of a compact set…

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Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

Q: Let C be a nonempty subset of a bounded set D. Why does inf C and sup C exist? Show that (c) inf D ≤…

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Q: let to be anonempty subset of abounded set E-Show that infESinf to Ssup E ssup E

A: Given :  E0 is a non empty subset of a bounded set E. To show :  inf E ≤ inf E0 ≤sup E0 ≤ sup E

Q: 5. Suppose that A and B are bounded subsets of R. Prove that AUB is bounded and that sup(AUB) =…

A: Given that A and B are bounded subsets of R.   We have to Prove that AUB is bounded and that…

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

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Q: Finish labeling the number of elements in the regions in the Venn diagram shown to the right, where…

A: Total number of elements in universal set, n(U)=100 Total number of elements in subset A, n(A)=30…

Q: Prove the following. 1. Bolzano-Weierstrass Theorem: If a bounded set S in R" contains infinitely…

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

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Q: hade in the region(s) that represent the set (AU B)nC' В A

A: Solution of the problem as follows

Q: 8. Prove that a matroid is uniform if and only if every circuit meets every cocircuit.

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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: let A Set bounded belou, erove that C. inf A Inf C A ン

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Q: let X={a,b,c} Construct two topologies for the set that have at least 4 open and that are…

A: Given that X=a,b,c. To construct two topologies that have at least four open sets. We know that in…

Q: e. If x is a subset of V of cardinality 2, then f (x) is the edge connecting the two vertices in x.

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Q: Define the sets D = In E N} n+1 Prove whether D is finite, countable, or uncountable?

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Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

Q: 3. If is a collection of dosed sets in R such that every finite subcollection has a nonempty…

A: Given : θ is a collection of closed sets in ℝ such that every finite subcollection has a non empty…

Q: Let A be a subset of R and let c€ R. Explain what it means to say that c is a cluster point of A.

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Q: (P,<) is a nonempty partially ordered set such that every chain in P has an upper bound, then P has…

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