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

Q: rove the theor

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Q: Let A be a set with |A| = n. Using induction, prove that |P(A)| = 2^n .

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Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

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Q: 7. Show that if S x T is a closed subset of R?, then S and T are closed subsets of R.

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Q: Prove that the set {a E R: a2 not equal to 6} is open.

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A: The given question is solved below.

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Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

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Q: Let T and M be a non-empty families of sets with T  ⊆ M. Show that ∩T  ⊆ ∪ M.

A: Given: T and M are families of sets with T⊆M. To show: ∩T ⊆∪M

Q: The Cantor set is closed .A O dense .B O open .C O neither open nor closed.D O

A: Closed

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: prove that the outer measure of Cantor Set C (x), 0<x< 1, is 1-x the generalized

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Q: 3. Let M and N be nonempty set such that M = N, then |M| = |N|. %3D

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Q: (3) Let S = {; Find with prove :m eN,n E N}. %3D suprenum of S. (4) What is an unbounded set?

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Q: For nonempty sets A, B, and C, let f: A > B and g : B >C. Prove: If g o f is one-to-one, then fis…

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Q: let to be anonempty subset of abounded set E-Show that infESinf to Ssup E ssup E

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Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

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Q: Show that for any nonempty set A, there exists unique set B such that for any x. xEB ex belongs to…

A: From your question, it is clear that "A" is a set of sets, i.e., members of the set "A" are sets.

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Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

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Q: For nonempty sets A, B, and C, let f : A –> B and g : B –> C. Prove: If go f is one-to-one, then fis…

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Q: For nonempty sets A, B, and C, let f: A -> B and g: B-> C. If g is onto, then gof is onto.

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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: Define the sets D = In E N} n+1 Prove whether D is finite, countable, or uncountable?

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Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup up A exist? Show that (a) inf…

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Q: Prove that for every n E N if A is a set with n elements, then the power set of A, P(A), has 2"…

A: The set P(A) contains every possible subset of A, including the empty set and the set A itself.

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)…

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 {$,}nEN is a disjoint seqeuence of measurable sets , then: 00 00 n=1

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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…

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Q: - A non-empty set E in R is connected if and only if it is an interval

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Q: 6. Let S be a set with a finite number of elements, and let f: S→→ S be a map (a) If fis onto, can f…

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Q: Let F and G be a non-empty families of sets with SG. Show that ng s UG.

A: Given: F and G are two non-empty family of sets with F⊆G, To show: ∩F⊆∪G

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”.