Let A CR be a nonempty subset which is boundedbelow, and x E R<0. Show that the scaled set xA isbounded above, that its supremum in R exists andsatisfies sup(xA) = xinf(A).

Given: A⊂R is a non-empty subset which is bounded below, and x∈R<0. We need to prove that scaled…

Answered: Let A CR be a nonempty subset which is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A be a non-empty set and suppose that f : A → Rand g : A →R. Define f + g : A → R by ( f + g)(x) = f(x) + g(x) Provide an example that shows sup(f +g)(A) < sup f(A) + sup g(A) holds, with f (A) and g(A) both bounded above.
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B₂ {(₂1 X12 X22 | Xij ≥ 0 for i, j = 1, 2, X11 + 12 = 2, X21 + x22 = 2, X11 + x21 (6) Let B₂ to be the set described in the screenshot attached Suppose Cij are real numbers such that C11C22 C12 + C21 > 0. Consider the following LPP: maximize: C11x11 + C12x12 + C21%21+ C22X22 subject to: (#11 #12) € B₂ X21 = 1.5, x12 + x22 = 2. (a) Write this problem in canonical form {x = (x11, 12, X21, X22) € R4 | Ax = b, x ≥ 0}. Hint: A should be a 3 x 4 matrix (after you eliminate one row which is linearly dependent on the others). (b) List all BFS. Which BFS x* has the maximum cost for this LPP? 2.5}. For the maximizer x*, show that for all basic directions D₁ at x* the reduced cost Cj = c² Dj ≤ 0.

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Let A CR be a nonempty subset which is bounded below, and x E R<0. Show that the scaled set xA is bounded above, that its supremum in R exists and satisfies sup(xA) = xinf(A).