Show that S CR is a closed set if and only if S contains all of its limit points.

Q: Show that S CR is a closed set if and only if S contains all of its limit points.

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

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Q: Prove that a point x is a limit point of a set A if and only if every e-neighborhood of x intersects…

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Q: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

Q: (b). Compute:  (i) Limit points. (ii) Interior of the set (iii) Closure of the set (iv) Boundary…

A: In the given question we have to calculate the Limit points , interior point ,closure , boundary…

Q: Show for every nonempty, finite set E that supE = maxE.

A: Given, E is a finite set. E is a non-empty set. To prove, supE = maxE We know that, sup is supremum…

Q: let A be a bounded. Sup A non. n-empty set in R show that is a limit point of A.

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Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

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Q: Give an example of a set w-limit that is a minimal set.

A: To give an example of a set w-limit that is a minimal set.  Given a collection of sets , a member…

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A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Prove (sn) is bounded if and only if lim sup |Sn| < +∞.

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Q: Write down the following, if they exist (a) all minimal elements (b) all maximal elements (c) all…

A: Maximal: If in a POSET an element is not related to any other element. Minimal: If in a POSET no…

Q: Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

A: Suppose a set M is closed and bounded. Then we have to show that M has a maximal and minimal…

Q: . Let 7 = {rn}~ be a sequence of real numbers. Prove that the limitpoint set of T is a closed set.

A: Given that x¯ = xnn=1∞ is a sequence of real numbers . We have to prove that the limit point set of…

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

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A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: Suppose A and B are both countably infinite sets. Show that the union of A and B is also countably…

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Q: ] Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

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

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Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

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Q: Prove the following analogue of Theorem 3.10(b): If {Ein} is a sequence of closed nonempty and…

A: given,En is a sequence of closed nonempty and bounded sets in a complete metric space X. En⊃En+1 and…

Q: Determine the interior, closure, and limit points of each of the following sets. (a) A= (0,1] in the…

A: As per expert guideline I am solving first question. Please post remaining questions separately.…

Q: Prove that if S is a bounded set then S is bounded.

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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: 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: be an open subset in C and f

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

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Q: Every nonempty class C has an E-minimal element.

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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: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

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: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: 2. Let E C R. Prove K is compact if and only if every infinite subset of K contains a limit point of…

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Q: g) for any bounded

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Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: Set A is bounded below if and only if set -A is bounded above. True False

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Q: A set L cR is convex if:

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Q: Find an example of on a set X (G has or H of G which has le

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Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

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Q: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

Q: In a bag of 350 chocolate candies, 37 of them are brown. The candy company claims that 13% of its…

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Q: Prove that every set with positive Lebesgue measure contain a non-measurable set

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Q: Prove that a set A is connected iff A is and interval.

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Q: Prove that set of real numbers E is closed and bounded if and only if every infnite subset of E has…

A: Consider the provided question, My objective is to prove that, set of real numbers E is closed and…

Q: A finite set .A is open .B has no limit point .C has a limit point .D uncountable

A: Option (B) is correct.

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.