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

Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v E R": Use some reasoning, or do some…

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Q: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

A: Let X be any infinite set with co-finite topology. Definition:  Let X be any infinite set. The…

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Q: Thoorem: is not closed set

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Q: i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii.…

A: Definition: A topological space X, τ is said to be Hausdorff if for every x, y∈X there exits U, V∈τ…

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Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: 2. Let X, Y be nonempty sets of real numbers that are bounded above. Let X + Y = {x + y:x € X and y…

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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: Queston2 {e-x, xe-O onlo, c∞) a) shau that q fandamented Set

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Q: Prove and show complete solution: Let ƒ be defined on an open interval (a, b) and assume that for…

A: please comment if you need any clarfification. If you find my answer useful put thumbs up. Thank…

Q: Let B= {x €R|x E Q} be a set. Is B countable? Justify the answer by showing the explicit formula for…

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Q: A set X and a topology on X which is not Hausdorff.

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Q: Exercise 5.2 | Show complete proof/solution. Let r > 0 and x E R". Show that the following set is…

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Q: Find an example of a nonempty set satisfying: 1. Closed, Dense in R 2. Open, Dense in R ~1. & 2.…

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Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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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 and B be nonempty sets of real numbers. If sup A exists then: a. sup (A U B) exists if sup(B)…

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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: An operation * is defined on the set R by x * y = x +y+ 6 where x, y are elements of R. What is…

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Q: Bounded Julia (ExH). Why is no Julia Set unbounded?

A: 37)

Q: (1) Construct an example for three distinguish points A, B and C in a 3- space for validating the…

A: Note: As per Bartleby’s guideline for more than one questions only first is to be answered please…

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

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

A: We have to prove this given statement

Q: In the Triangle Inequality of Euclidean n-space: Prove ||x - z || <= ||x - y|| + ||y - z ||…

A: The triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle…

Q: X, Y and Z be subsets of the set . Verify the following properties: XCY XUY=Y⇒XnY=X -UI 3)

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Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v € R": Use some reasoning, or do some…

A: We have to prove ∑i=1nuivi≤∑i=1nui2∑j=1nvi2 Since we know the Holder's inequality…

Q: Prove the empty set hasn't supremum and infimum

A: Lower Bound: A lower bound of a subset A of a partially ordered set (X,⩽) is an element a∈X such…

Q: Đ A is a closed set.

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

Q: (ii) Find a connected subset of R² which is not convex.

A: Note: Hi! Thank you for your question. As per the guideline we can solve one question at a time in…

Q: Let 5- {-3, -1, 0,1 V5,4,3). Determine which elements of S satisfy the inequality. (Enter your…

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Q: Find an example of a nonempty set satisfying: *Not Open, Not Closed, Dense in R *Open, nowhere…

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Q: Let A = {x ER|x' E Q} be a set. Is countable? Justify the answer by showing the explicit formula for…

A: Please check the answer in the next step

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

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: Proof that the subset {(x, y)/x # 0, and y <-1} is an open set in R?-

A: Let S=x,y| x≠0, and y&lt;-1.  A subset S⊂ℝ2 is said to be an open subset, if its complement is…

Q: Using the definition of an open set, prove that the interval (0, 1) is an open set.

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Q: Let A, B be disjoint infinite sets. Prove that their union A U B is denumerable if and only if both…

A: Given A , B be disjoint infinite sets. Firstly, consider A∪B is denumerable Since, A⊆A∪BandB⊆A∪B…

Q: X - (Y UZ) = (X – Y)nZ°

A: Assume that X, Y and Z are sets. Prove that, X-Y∪Z=X-Y∩ZC. That is, a∈X-Y∪Z⇒a∈X-Y∩ZC  and…

Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B)n (A × C)

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A: Let, s be the circle given by, s=x,y∈ℝ2 :x2+y2=2  Then s is a closed subset of ℝ2. Let us consider a…

Q: Show That R is Banach Sface.

A: To Show -

Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

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Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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

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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 if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

Q: The set of all triples of real numbers (x, y, z) with the operations x, y,2) + (x', y',z') = (x +…

A: Given : set of all triples of real numbers x, y, z with the operations   x, y, z+x', y', z'=x+x',…