Let A be a set and d : A × A → {0, 1, 2} a symmetric set function that satisfies d(x, y) = 0 ⇔ x = y. Show that (A, d) is a metric space.

Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v E R": Use some reasoning, or do some…

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Q: Prove the Triangle Inequality |v+w| <= |v| + |w|  (Hint: Use Cauchy Schwartz)

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Q: State whether each of the following sets is open, closed, or neither. Which are compact? (You do not…

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Q: A PARABLE OF THE FORM ar²+bx+c WITH <0 ALWAYS a) has an absolute maximum b) has an absolute minimum…

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Q: (a) Find the supremum of the set S:= 4 13 40 13 9 27 81.. }, if the supremum exists. Justify your…

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Q: (1) Write the definition of a measurable set ECR. (2) By writing the definition of the measurability…

A: Let m* be the outer measure of the power set of the set ℝ. A subset E⊂ℝ is said to be measurable if…

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Q: 1) (b) (c) Consider the set X = {x € R¹ / 93x1 +34x2 + 26x3 + 18x4 ≤ 131, x1, X2, X3, X4 € {0, 1}}.…

A: Given: Set, X=x∈ℝ4/93x1+34x2+26x3+18x4≤131, x1, x2, x3, x4∈0, 1We need to show that the valid…

Q: a) ||x|| = (x1)² + (x2)² (Euclidian norm) b) ||x|| = (x1)² + 5(x2)² c) ||x|| = |x4| + |x2|

A: As per the company rule, we are supposed to solve the first three subparts of a multi-parts problem.…

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Q: Theorem 1.10. A set is infinite if and only if there is an injection from the set into a proper…

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Q: tify rigorously the given set is closed in R³. A = {(x, y, z) | x ≥ 0, y ≥ 0, z ≥ 0, x+y+z ≤ 12}. B…

A: We can prove by showing complement of a set is open.

Q: 5. a) Let V₁ = H b) Let V₁ = 11 ----- = V3 Does (V1, V2, V3} span R¹? Why or why not? Does (V₁, V2,…

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Q: (e) A x (B – C) = (A × B), (Ax C). %3D (f) (A – B) × C = (A x C). (Bx C). (g) If AC B, then A x C ВХ…

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Q: 3. Determine whether the following sets are open, closed, both, or neither. (a) (2,3) U{4} (b) [2,3]…

A: (a) (b)

Q: Describe the following sets by roster method: Example (i) {x : x + 5x + 6 = 0, x e N} (iii) {x : 4x…

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Q: 12. A path is said to be closed if the function that generates it satisfies f(0) = f(1). Show that…

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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)|&lt;(π/2)} is:   Closed: The set includes all complex numbers…

Q: C[[a,b],R") = {f : [a,b] → R", f=(f,...,f)/ f is continous on [a,b]} Show that the following…

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Q: 7. Sketch the set {(x, y) = R² : ||(x, y) || = 1} when: (a) ||(x, y)=√√√x² + y² (b) ||(x, y) || =…

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Q: (1) Construct an example for three distinguish points A, B and C in a 3- space for validating the…

A: For every three-dimensional vector a→=ai^+bj^+ck^, the corresponding point has the coordinate a,b,c.…

Q: 6. Are the statements true or false? Prove or give a counter example. (i) (3æ)(Vy)(x + y = 2).…

A: Hello, thanks for your question but according to our policy I am answering the first three subparts.…

Q: 2 and v = [*] 21. Let u = Show that is in Span {u, v} for all h and k.

A: We have to solve given problem:

Q: b) Prove that 3abc ≤ a³ + b³ + c³ for nonnegative a, b, c. (Hint: In the in- equality of part (a),…

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Q: Exercise 4.6. Decide in each of the examples what kind of ordered set it is, poset, strict poset,…

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Q: Let R = Zg, and let N = R³. Determine whether or not X = {(4, 5, 4), (4, 5, 2), (2, 2, 3)} is a free…

A: It is given that,  R = ℤ8  and  N = R3. We have to determine that whether the set,  X = 4, 5, 4, 4,…

Q: Prove that if the joint caf of X and 4 satisfies". Fx₁y (x,y) = Fx (x) Fy (y) ху then for any pair…

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Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

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Q: Show that each of the following functions is convex or strictly convex on the specified convex set:…

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Q: Please answer the whole question. It's just true/false

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Q: Investigate the following func Mon por Local maximun and Local minimum point. 3) RCX, Y, 2 ) = x² +…

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Q: Justify rigorously the given set is closed in R³. (a) A = {(x, y, z) | x ≥ 0, y ≥ 0, z≥0, x+y+z≤…

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Q: Find a basis for the null space Nul(A) and a basis for the column space Col(A) of the 2 6 -10 -4 -17…

A: Use basis concept

Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v € R": Use some reasoning, or do some…

A: We have to prove ∑i=1nuivi≤∑i=1nui2∑j=1nvi2 Since we know the Holder's inequality…

Q: Determine whether the following sets are convex (1) {(x,y)=√x + √√]]l}

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Q: A and B be two sets, each one of them having a minimum. Assume that A n B is not ty. Then An B has a…

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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…