Use double set inclusion to prove that dep. PANB) = P(A)n P(B).

Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and x,…

Q: If ftickiieI{ is acollechion of Subsets ofX then く) andh %3D det the equality not (X;)-( true in…

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Q: Show that R has the same cardinality as the interval (–2, 2) by building a bijection between the two…

A: This is a problem of Real Analysis and Topology.

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: 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: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: 2. if it was P(x)=ao+a,%tazk Prove that PecR) inner multiPlication sPace

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Q: (ii). Prove that a set S CR is everywhere dense iff it intersects every non-empty open sets in…

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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: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

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Q: Let A be a denumer able set. Prove that A has a denumer able subset B such that A - B is…

A: Let A =set of multiples of 4, B=set of multiples of 3. Both are denumerable. Intersection of A and B…

Q: prove tuat Only Ø ond IR ere open Subsets of IR Mice handwritting will be appreciated!

A: In the given question we have to prove or disprove that ∅ and ℝ the only open subset of ℝ. Note: the…

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

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Q: Prove that every denumerable set A can be partitioned into a denumerable number of denumerable…

A: In many books you see different kinds of proof , but I can explain you it easily by "pigeonhole…

Q: prove the ITT₂TT, ITKITI"

A:

Q: rsection of any collection of com

A: This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the…

Q: 4) Using equality of sets, prove that (An B)' = A' U B'

A: we know thatA∪B=x: x∈A or x∈BA∩B=x: x∈A or x∈BA'=x:x∈U, x∉A=U \ AWhere U is a universal set.That is…

Q: Every connected subset of R is complete is ture or false?

A: Every connected subset of R is complete is true or false?

Q: Is the Cantor set closed? Is it compact? Explain!

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Q: Classify the discontinuity of fas a removable, jump, or infinite.

A: There are three types of discontinuity: 1. Removable Discontinuity: In this type of discontinuity,…

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: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: (ii). Prove that a set SCRis everywhere dense iff it intersects every non-empty open sets in R.Prove…

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Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

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

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Q: umat is bounded, and is not compact in Q.

A: Here the closed means closed with respect to subspace topology. And compact means every cover has…

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

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Q: 3.) Prove that P, is a all ve ctor by showing space 10 0xioms ore

A: proof P2=a0+a1x+a2x2:a0,a1,a2∈R   is vector space over + and ×first we proof P2 is abelian group…

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: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: Show that any compact set in R is closed and bounded.

A: Proof

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

A:

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

A: Schroder-Bernstein Theorem: If there exists injective functions f : A→B and g : B→A between the sets…

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: 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: 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: Explain the idea of cardinality. When is a set denumerable? Show that the union of two denumerable…

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Q: f CB) is aelesed Sot at y for eVery closet Set Bat X Prove that fEÃ) cFA)YACX then SupPose CA) <…

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Q: open ball is the open set

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Q: Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.

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Q: a nonempty compact set of IR. sup Ke K. be

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Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

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Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

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

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Q: Let X = {a, b, c} and R={(c, b),(a, c)}. Then Check the all properties reflexive,symmetric,…

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