9.K Let X = Q y K = {p € Q : 2 < p? < 3} . Show that K is closed in Q, is bounded, and is not compact in Q.
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- Suppose poset (A, R), where A = {1, 2, 3, 4, 6, 8. 9. 12, 16, 18, 32, 36} and the relation R runs outdivide. Answer the following questions:a. Draw a Hasse Diagram for the poset (A, R)b. Find the upper limit of {6, 9}c. Supremum of {6, 9}6. Let A = {x|x ∈ R and x^2 − 4x + 3 < 0} and B = {x|x ∈ R and 0 < x < 6}. Prove that A ⊆ B.of timization : Egioobl TQ/ Find the minimum setetion Point (interms frshamif MSing the LagrantTemethot for the constraftmedoftimiZation Prablem where frofermA B aremstnt Vatues in A/fot 20/ Let (xy*)=(os nimum foint forthe restrictet eptimization frobltem foin Find the Valueofß Phat makesthe Soletion foint (x" 1)a. that Saetisfies the necessary and Sufficient haj rang Condtition forthe existoncef alocal minimum foint
- If S is a non-empty subset of R which is bounded below, then a real number t is the infimunm of S iff the followving two conditions hold : (i) x2t V xeS. (ii) Given any ɛ> 0, 3 some xe S such that xLet A = (0, 1]-{|n € N}. (a) Find the set Aº of its interior points. (b) Find the set A' of its limit points. (c) Determine whether A is open or closed. (Discuss both.) (d) Determine whether A is compact. (e) Determine whether A is connected. (f) Find the set 8A of its boundary points.Exercise 1. A bucket contains 10 ping pong balls, of which 3 are orange and 7 are blue. You draw two ping pong balls out of the bag (randomly and without replacement). Let X be the number of blue balls that you draw. (a) Explain why X meets the criteria to be a discrete r.v., and give its space. (b) Find the pmf of X.solve quick(a) (b) S = 5 12 (0·0·000} 1 Is S a spanning set for R¹? Justify your answer. 10 7 Is S linearly independent? Justify your answer.Suppose X∼t(10)X∼t(10). Which of the following code gives you the central limits aa and bb such that P(a<X<b)=0.95P(a<X<b)=0.95? A. a <- qt(0.025, df = 10)b <- qt(0.025, df = 10, lower.tail = F) B. a <- qt(0.975, df = 10, lower.tail = F)b <- qt(0.975, df = 10) C. a <- qt(0.025, df = 10 , lower.tail = F)b <- qt(0.025, df = 10) D. a <- qt(0.975, df = 10)b <- qt(0.975, df = 10, lower.tail = F) E. A and B F. A and D G. B and C H. C and DQ1: Find Sup ; Inf; Max; Min for the following sets: m { e z*} . (a) S = 2n : m,n E Z* }; (b) T = {n+1 :n E Z+ Q2: (a) Let a ,b E R, and a < b. Prove that 3s ER- Q, a11B₂ {(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.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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