27Prove that if A is not connected then A is not an interval.

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?

A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…

Q: A string of Lcoincides with the interval 0, L

A: Given, A string coincides with the interval on the . We have to set up the boundary value problem…

Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(A U B)…

A: Let A,B not empty, bounded subsets of R .Prove thatsup(A∪B)=max{sup(A),sup(B)}. Let: ⇒x∈A∪B⇒x∈A or…

Q: Let A be a nonempty subset of positive integers and B = {n + v5 : n E A}. Show that B has a least…

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Q: 3. Find an example of two nonempty sets A and B for which A x B = Bx A is true.

A: To find an example of two nonempty sets A and B for which AXB=BXA

Q: .

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Q: Let S={(a,b) : a, b \in Q} be the collection of open intervals in R with rational endpoints. Find…

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Q: 3) Let A be a nonempty set of real numbers that is bounded below. Let -A be the set of all numbers…

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Q: (12) Give an example of a domain X and predicates p() and q() such that Vx € X, (p(x) → q(x)) is not…

A: Step:-1 Since, we know  P→Q is true if  statement P is true and Q is true or P is false and Q is…

Q: 5. Let S be a set of negative numbers. Prove the least upper bound of S cannot be positive.

A: To show that the least upper bound of a set of negative numbers cannot be positive.

Q: Prove that if ƒ = {-2 if x is irrational , then f € R on [a, b].

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Q: 4. Let R be the relation in the positive integers

A: 4. Given, R be the relation in the  positive integers ℕ defined by                        R=x, y: x,…

Q: 4) State the condition for equivalnce for measurable set. being a lebesgue

A: Disclaimer: Since you have asked multiple question, we will solve the first question for you. If you…

Q: E is closed. - E is complete, i.

A: It is given that X is a complete metric space and E is a subset of X. We have to prove that the…

Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: Let A be a non-empty, bounded set of positive real numbers. Let B be the set consisting of numbers…

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Q: Let S be any set of non-negative integers with k + 1 elements. Show that there is x, Y E S s.t. k…

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

A: let A and B be nonempty subsets of reals.

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: Let R (the set of real numbers) be equipped with the discrete topology, then the semi open interval…

A: Let (X, d) be a discrete metric space. Thus singletons are open sets as {x} = B(x, ϵ) where ϵ <…

Q: 12. The relation R defined on the set of integers by a R b means 'a + b2 3' is A. Reflexive B.…

A: As per our guidelines, we are allowed to answer only one question. Since you have asked multiple…

Q: Let a and b be positive integers such that ax + by = 7 for some integers x and y. Then %3D

A: We have to choose  current answer of given statement.

Q: 5. Prove that if ƒ : [-1, 1] → R is measurable, then |f| is also measurable.

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Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: (i) Show that a non-empty subset of real numbers A is nowhere-dense if and only if the interior Aº…

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Q: A set consisting of single point has measure zero .A O nonzero measure .B O open .C O none of…

A: We have to find the measure

Q: Let S be a nonempty set of real numbers that is bounded from above and let x = sup S. Prove that…

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Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: If A CB…

A: If you understand the concept of inf and sup then these statements are obvious. Remember that inf of…

Q: Đ A is a closed set.

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

Q: Let [a,b] be a closed interval which is by a class 9 of open intervals. Prove the contains a finfite…

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Q: Let a1, a2, a3, . .., an be real numbers. Prove that at least one of them is greater than the…

A: We prove it by contradiction.

Q: Prove or disprove: All uncountable measurable subsets of ℝ have positive measure.

A: All uncountable measurable subsets of ℝ have positive measure.

Q: let 19, b) = 1, them prove that there exists x20 Such that ijzo ax_by = 1

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Q: Prove that if A is not connected, then A is not an interval.

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Q: let a∈R. give an example of nested intervals of each of positive length such that the intersection…

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Q: a. Let S= Z (the set of integers). Define t < s by s – t is non-negative. Prove or disprove that…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Find the concavity intervals of Cari selang-selang kecekungan bagi f(x) = 2x³ – 9x² + 12x – 4

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Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(AB) =…

A: Suppose A and B are bounded and nonempty subsets of real numbers and λ∈R. We need to prove that…

Q: 6. Prove that |x.y| = |x|.|y|, where x and y are real numbers.

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Q: 5. Differentiate with respect to x: SET A. 5e

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Q: Two runners start a race at the same time and finish in a tie.Prove that at some time during the…

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Q: Is this set finite or infinite? {x € R|1 < x < 2}

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Q: Assume that ∀x∃yP(x, y) is false and the domain of them is nonempty. Which of the following   must…

A: See the detailed solution below.

Q: Let A be a nonempty subset of positive integers and B = {n + v5_ : nE A}. Show that B has a least…

A: We will use the well ordering principle to prove that B has a least element.

Q: Let n and r be non-negative integers such that r <n. Then, C + "C , = **C,

A: nCr=n!r!(n-r)!

Q: 5. Determine whether the set of all ordered triples of real numbers with the op- erations defined…

A: Given: (x,y,z)+(x',y',z')=(x+x',y+y',z+z') k(x,y,z)=(x,1,z)

Q: 3. Prove if x is an isolated point of a set S, then x E bd S.

A: Since you have posted a multiple question according to guildlines I will solve first question(Q3)…