10. Logical. Given a set in 0-1 variablesnX = {x € B" :> a;x; < b}j=1with a; 20 for j = 1,..., n, under what conditions is(i) the set X empty?(ii) the constraint -1 ajx; < b redundant?(iii) the constraint x;(iv) the constraint x; + x; <1 valid?= 0 valid?%3D

As per company guidelines we are allowed to solve maximum three sub-parts so i am solving first…

Answered: 10. Logical. Given a set in 0-1…

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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VIII.1. Let A = {1,2, 3, 4} and let R, S, T and U be the following relations: R= {(1,3), (3, 2), (2, 1), (4, 4)}, S = {(2,1), (3, 3), (4, 2)}, T = {(4,1), (4, 2), (3, 1), (3, 2), (1, 2)}, U = {(x, y) | x > y}. (a) For each of R, S, T and U determine whether they are functional, reflexive, symmetric, anti-symmetric or transitive. Explain your answer in each case, showing why your answer is correct. (b) What is the transitive closure of R? (c) Explain why R", the transitive closure of R, is an equivalence relation. De- scribe the equivalence classes E, into which the relation partitions the set A. VIII.2. Prove or give a counterexample to the following statement: for any relation R both R and RoR always have the same transitive closure. VIII.3. Is there a mistake in the following proof that any transitive and symmetric relation Ris reflexive? If so, what is it? Let a Rb. By symmetry, bRa. By transitivity, if aRb and bRa, then aRa. This proves reflexivity. VIII.4. Determine for the…

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10. Logical. Given a set in 0-1 variables n X = {x = Bajxj ≤ b } j=1 with aj 20 for j = 1,..., n, under what conditions is (i) the set X empty? j=1 (ii) the constraint -1 ajj ≤ b redundant? (iii) the constraint x; = 0 valid? (iv) the constraint xi + xj ≤ 1 valid?