(a) Show that a set S in R" is convex if and only if for each integer k, every convex combination of any k points of S is in S.

Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: e) Prove the theorem that states that “S" is a countable set, and there exist a surjection of N onto…

A: Two sets which are one-one and onto, have same cardinality. Let S is a countable set,  S=a1,…

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

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

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: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: For any collection a of subsets of X, there i ntaining a.

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Q: Prove that intersection of any class of convex subsets of N is convex?

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Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…

A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…

Q: (c).Prove that set of real numbers E is closed and bounded if and only if every infnite subset of EE…

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

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…

Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

A:

Q: Let X be an ordered set, and let A be a subset of X.Consider A as an ordered set, where the order on…

A: Given that X be an ordered set and let A⊂X, A is an ordered set. Y⊂X and Y is convex  in X. The…

Q: (a) Prove that if A and B are nonempty, bounded sets of real numbers such that x ≤ y for all x € A…

A: a) It is given that A and B are nonempty, bounded sets of real numbers such that x≤y for all x∈A and…

Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

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Q: Show that if Sı and S2 are convex sets in R"+n, then so is their partial sum S = {(x, yı + y2) | x €…

A: We will solve the question number 2.16 as mentioned. Given that S1,S2 are two convex sets in ℝn+m.…

Q: Let X and Y be two discrete spaces, then * O X is never homeomorphic to Y X is homomorphic to Y if…

A: option (c) is correct.

Q: the book is "First Course on Fuzzy Theory and Applications"

A: To prove: All the α–cuts of any fuzzy set A defined on Rn­ are convex if and only if

Q: Every bounded domain subset of D containing

A: Set theory

Q: Prove that if ø is convex, then a-[$(x + Az) – Þ(x)] is nondecreasing for å > 0.

A: As  φ is convex we conclude that its second derivative :  φ'' is always positive ∴  φ''(x) > 0…

Q: Suppose that N+ 1 balls, of which a+b+1 are red and N-a-bare blue, are arranged in a linear order in…

A: From the given information, There are N+1 balls, of which a+b+1 are red and N-a-b are blue arranged…

Q: If S and T are nonempty bounded subsets of the real numbers, show that if ScT Then infT < inf S <…

A: Given ,  S is subset of T

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find the order of Z_2 x Z_2.

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…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Let X be an n-element set. Prove that there are 3n ordered pairs (A, B) of subsets of X such that A…

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Q: There is a minimum integer r such that for all integers k >= r, there exists x,y such that k = 5x +…

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Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

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Q: Prove or disprove whether or not the following two-dimensional set is convex.  S = { (x1,x2 ) such…

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Q: 7. Let E and 0 the set of even and odd integers, respectively. Prove that {E, O} partitions Z.

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Q: Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…

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Q: For any measurable nonempty subset of R, L∞ (N) = I¹ (N).

A: Given: Measurable nonempty subset Ω of ℝ. To prove: For any measurable nonempty subset Ω of ℝ,…

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: In a discrete space X * O the only dense subset of X is X O every subset of X is dense in X None of…

A: Dear Bartleby student according to our guidelines we can answer only three subparts, or first…

Q: Let X and Y be two discrete spaces, then* O x is never homeomorphic to Y O X is homomorphic to Y if…

A: As two discrete spaces are homeomorphic iff they have same cardinality .

Q: Prove that if R, S, and T are nonempty sets, then R×T ⊆ S×T if and only if R ⊆ S.

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Q: A fuzzy set B is closed if and only if B contains all the accumulation point of B. accumulation…

A: Option-A is correct.

Q: Prove the following. 1. Bolzano-Weierstrass Theorem: If a bounded set S in R" contains infinitely…

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

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Q: Let (X. T) be a T, topological space and A a nonempty subset of X, if x is a cluster point of A…

A: Given: Let (X,T) be a T1 topological space and A a non-empty subset of X, if x is a cluster point of…

Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: 12. If E is a bounded above subset of Z, then A. sup E exists and sup E E E B. sup E exists and sup…

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Q: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

A: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

Q: e. If x is a subset of V of cardinality 2, then f (x) is the edge connecting the two vertices in x.

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Q: (a). Find a set S and a bijection between S and a proper subset of S. (b). What condition on S…

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Q: A subset S of IR" is convex if whenever it contains a and b it also contains the line segment…

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

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Q: In a discrete space X O every subset of X is dense in X the only dense subset of X is X O a proper…

A: Thanks for the question :)And your upvote will be really appreciable ;)

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…

A: