Which of the following subsets of G CC are (i) open, (ii) connected, (iii) a domain? (a) G = {z € C; |z| < 2} \ {z € C; Rez = 0}, (b) G = {z € C; |2| < 2} n{z € C; Re z > 0}, (c) G = {z € C; |2| < 2} U {z € C; Re z > 0}, (d) G = {z € C; |2| < 2} \ {z € C; Re z = 0}. %3D
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- Let 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
- Show the following set is convex. x10 Determine a-cut sets of the above set for a=0.5, 0.8 and 0.9.2. Find the infimum, the supremum, the minimum and the maximum (if they exist) of the following sets: (i) A = {1,3,5,7.9,11} (-1)" (iii) A 4. 2 3 4 (v) A={x eR:-4<2} 3. Which of the sets in (2) are bounded?Determine the accumulation points of the set {z : -π<Arg z<π}
- 4. Let X = {n €N : 10Let x = (x1, x2, x3) ER³ and suppose you are given a subset UCR³ defined by U = {ER³ : x₂ < 5} Which of the following statements are necessarily true? (i) The complement of U contains a boundary point. (ii) The complement of U contains no interior points. (iii) U is not an open set. (iv) U is not a closed set. (A) (iii) only (B) (i) and (iv) (C) (ii) only (D) (i), (iii), and (iv) (E) (ii) and (iii) (F) (iv) only (G) (i) only (H) (iii) and (iv)1. Let S = (0,7) U 27 +13... Define < on S by a4B₂ {(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 ≤ 0.6.1.8 (a) Suppose that |A|| = 3, and |B| = 4. What are the minimum and maximum values for the cardinalities |(A × B) N (B × A)| and |(A × B) U (B × A)|? (b) More generally, suppose that |A|| = m, |B| = n and |AN B|| = c. What are the above cardinalities?Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,