Show the following: (a) (b) If f is K-convex, then its sublevel sets Ca are convex for all a. f is K-convex if and only if epik f is a convex set.

Q: Let A be an arbitrary set. Show that there does not exist a bijective function f: A → P(A). = Hint:…

A: As per the guideline, I am supposed to solve only first question from the given problems.

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: Use the Heine-Borel Property from (5.1), which states that a set of real numbers is compact if and…

A: See solution below. Here we used definition of bounded and closed sets.

Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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Q: Prove the following for any given sets A, B, C and D. P(0) = {0}

A: Ø = null/empty set.  [A set containing no elements]   Now, for any set A, P(A) denotes the power set…

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

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Q: For nonempty sets A, B, and C, let ƒ : A —> B and g: BC. Prove: If g of is one-to-one, then f is…

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Q: Let A, B, C be nonempty sets and let f : AB, g: B → C be functions such that go f is bijective.…

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Consider the following sets in the dictionarv order. Determine which of these sets are linear…

A: We know that Linear continuum is a linearly ordered set S of more than one element that is densely…

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

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Q: Q1\ Define the compact set. Then, prove that: if A is a compact subset of F , there exists a best…

A: Compact set : A set is compact if and only if every open cover has a finite subcover. There is also…

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

A: Let's suppose that A and B are 2 convex sets. We have to show that A∩B is also convex. Let's take…

Q: For each of the following functions f(x) (x E R²), describe the sets V₁ ≥ 4 = { x | f(x) ≥ 4 } and…

A: A function is called convex if its double derivative is positive OR any line formed by joining any…

Q: Prove that: m*(A) 2 0, for all sets A

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Q: 1. For each of the following sets, decide whether they are closed, bounded, compact (both) or…

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Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

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Q: Give the definition of a Lebesgue measurable set. Prove that if A and B are Lebesgue measurable…

A: We are given that A and B are Lebesgue measurable sets.        => m(A) < ∞,            m(B)…

Q: 3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are…

A: As per guidelines, I can do only first three subparts of the question.

Q: Express the characteristics of the following sets of points, in terms of: closed or open, connected…

A: The set of points {z∈C: |arg (z)|<(π/2)} is:   Closed: The set includes all complex numbers…

Q: - If C is Convex set ond a GR, then XC is a Convex set

A: Let C be a convex set and x∈R Then we have to show that xC is a convex set. The set xC is defined as…

Q: Which of the following statements is always true ? Select one: O a. If A, B, and C are sets with AC…

A: Recall countability and uncountability

Q: b Let A be a continuous fuzzy quantity. If all the a-cuts of A are closed intervals then show that A…

A: In the question provided it is given that A is a continuous fuzzy quantity. The objective is to show…

Q: 3. Use the definition of convex sets to answer the following: Show that if the sets S and T are…

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Q: Show that the set given in the image below is non convex. Please prove the result mathematically…

A: A set is said to be convex if for any two points in the set the line segment joining those points is…

Q: Let B be a set and let A ⊆ B be a subset. Suppose that there exists an injective function f : B → A.…

A: Schroder-Bernstein Theorem- Let A and B be arbitrary sets. Using the concept of cardinality. It is…

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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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: A4For any two finite sets A and B, define f(A,B) to be the number of elements that are contained…

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

A: A function is said to be one-to-one if and only if  implies  for (1) Prove by the direct method…

Q: For any set X with at least 2 elements, define a function f : P(X) x P(X) → P(X) by f(A,B) = A △B…

A: We have taken two different ordered pair of subsets of X and shown that their image is same. Hence f…

Q: If D(L AOB) states the set of points located on the OA side containing point B and the OB side…

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Q: Given two sets X and Y, we denote the cardinality of X by |X| and the set of all functions from X to…

A: We are given two sets X and Y. The set of all functions from X to Y is denoted by YX.  Suppose that…

Q: a. Prove that for any sets A,B⊆X and for any function f:X⟶Y, f(A∩B)⊆f(A)∩f(B).       b. Prove that…

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Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Let B = {((-1)n*n)/(n+1) : n = 1, 2, 3…}. Find the limit points of B. Is B a closed set? Is B an…

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Q: (a) Represent the set {x = (x1, x2) = R² | x1x2 ≥ 1}, as the intersection of some family of…

A: The given set is x=(x1,x2)∈R+2|x1x2≥1. To Find: a) Represent the set x=(x1,x2)∈R+2|x1x2≥1 as the…