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 # 0 and e1, €2 > 0. Prove that x is a convex combination of x1 and x2. That is, prove that x = ax1 + (1 – a)x2,

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

Q: Determine whether the set in P_2 is linearly independent.    S={x^2 , +1, 2 - 1, 2}

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Q: j) A U A = A k) A U $ = A

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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: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).

A: It is given that X,Y  be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…

Q: 1.4 Determine whether the following fuzzy sets are convex or not. a) A- Ju,(x)/x where ,(x) =1(1 +…

A: (a) A=∫μA(x)/x  where μA(x)=11+x2, -∞<x<∞ It is clear that z=λx+(1-λ)y, 0≤λ≤1 For x,y ∈ℝ…

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 two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

A:

Q: Let M1 = ( S1, Ih) and M2 = ( S2, I2) be two matroids with disjoint ground %3D sets. Prove that the…

A: Given: Two matroids M1=S1,I1 and M2=S2,I2 with disjoint ground sets.   To prove: The subset system…

Q: Suppose that |A| = m and |B| = n. Find the following cardinalities. a. b. Ip(A x B)| c. Ip(A) x…

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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: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

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

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Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: Exercise 6.65. Prove that if E₁ CR and E₂ C Rm are convex, then E = E₁ × E2 is a convex subset of…

A: Given :  E1⊂ℝn and E2⊂ℝm be any two convex sets. To show :  E = E1×E2 is a convex subset of ℝn+m…

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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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: 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: Find an example to show that the convexity of S is necessary in Exercise 19. Reference Exercise 19:…

A:

Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x,y) <r} Show that B(x) is closed set…

A: Given: τ=B⊆X : ∀x0∈B, ∃rx0&gt;0, such that Brx0x0⊂B is a topology in X. Open ball Br(x)=y∈X : d(x,…

Q: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement ofP, then: L(f;P) <…

A: Answer

Q: 3. Suppose that {v,,v,,v,} is a linearly independent subset of R". Show that the set {v,,v, +v3,v; +…

A: The set {v1,v2,......vn} is linearly independent if for the combination, av1+bv2+......nvn=0 for…

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: 5. Determine whether the set in P2 is linearly independent. S = {x2 + 1,2x – 1,2}

A: Given set,  S=x2+1,2x-1,2

Q: Prove that P(Sn T) P(S) n P(T) for any sets S and T

A: Let, S and T be a any set. also, P(S) be any power set of S and P(T) be any power set of T.x∈P(S∩T)…

Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: Show that E is a measurable set if and only if for every E > 0 there is a closed set F and an open…

A:

Q: (a) Suppose M is a set and d, d' are two different metrics on M. Prove that d and d' generate the…

A: Please find answer below:

Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

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Q: 18: Consider the two linear maps T, and T2 on V, defined as T(X, X, x) = (0, xz x) and T,(x , x, x)…

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Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: Let X be a nonempty set and fix x E X. Define (1, rE A (A) = 10, 4 A

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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: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

A: Let f: ℝn→ℝm be an affined map defined as x↦Ax+b, where A∈ℝm×n and b∈ℝm. Then it follows: fx=Ax+b…

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

A:

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: 9. Suppose u, v, w e R" form a linearly independent set. Prove that u + v, v + 2w, and -u + v+w…

A: Given that: u, v, and w form a linearly independent set. To prove that u+v, v+2w, and −u+v+w form a…

Q: Let A = {2, 4} and B = {1, 3, 5} and define rela- tions U, V, and W from A to B as follows: For…

A: (a) First obtain the 3 relations as follows.

Q: a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

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Q: If (X,, T,), (X, T2) are two topological spaces, and A,, A, are open sets in X,and X, respectively,…

A: