Let S be an ordered set. Let A ⊂ S be a nonempty susbset that is bounded above. Suppose A exist and sup A ∉ A. Show that A contains a countably infinite subset. Show A is infinite.

Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

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Q: Let A be a nonempty subset of R. If a = sup A is finite, show that for each ɛ > 0, there is an a E A…

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A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

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Q: 2. Let S be a bounded set in R and. let a < 0. and let aS := {as : s E S). Prove that inf(as) = a…

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Q: Let A be a non-empty subset ofR which is bounded below and let B be the subset of R defined by B =…

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Q: Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS

A: According to the given information, it is required to prove that:

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Q: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

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

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Q: Let W be a nonempty subset of R that is bounded above, and define the set B as follows B= {N ER:N is…

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Q: . Let S be a bounded set in R and let T be a nonempty subset of S. Show that inf S <inf T< sup T <…

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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: Let A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

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A: We can prove this by the definition of upper and lower bound

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A: Set theory

Q: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

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Q: Let A and B be bounded nonempty subsets of R, and let A + B:= {a +b: a E A, be B). Prove that sup(A…

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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: 6. Let A and B be bounded nonempty subsets of R, and let A+ B:= {a +b: a e A, be B). Prove that…

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Q: Prove that a set A is countable if and only if there exist a surjection from B onto A.

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Q: Let A, B Be two subsets af R, such that B is bounded and YaE A ]bEB such that show that: a sb O A is…

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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: Let A be a subset of some universe U. Prove that AN A° = Ø.

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Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: Let (E), be a countable disjoint collection of measurable sets. Prove that for any set A, 00 m* (AnU…

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Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

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Q: Prove that a set A is countable if and only if there exist an injection of A into N.

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Q: Label each of the following statements as either true or false. The least upper bound of a nonempty…

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Q: )Š is the smallest closed set in M such that S CS.

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Q: the collection of all close subsets of R is called the topology of R ylgn İhi

A: False

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Q: Let A be a nonempty subset of R that is bounded below. Define the set B={b: b is a lower bound of…

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Q: 2) let A be a finite set and a EA.Lf a EA is not a maximal element of A,then a has an îmmediate suc…

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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: Let S be a set with a finite number of elements, and let f: S - S be a map _ (a) If fis onto, can If…

A: We are given that S is a set having finite number of elements and f : S--&gt;S be a map.Before we go…

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.