Please solve problem 3.7 in detail please 3.1. Linear Order. Our goal is to define a topology on a set induced by an order on thatset. But first we have to make clear what we mean by “order" on a set.Definition 3.1. Let X be a set. A relation R on X is a set of ordered pairs from X × X.For x, y E X, rather than write (x, y) Erelation between x and y.we usually write xRy, putting the symbol of theExercise 3.2. Consider the relation "is a subset of" symbolized C on the collection of allsubsets of N. Rather than write ({1,2}, {1, 2, 3}) €C, we write {1,2} C {1,2, 3}. For thisrelation, determine the answer to each of the following questions:(1) Given A, B subsets of N, are they always related by C?(2) Given A, B subsets of N, is it possible that both A C B and BC A?(3) Given A, В, С subsets ofN, if A c В and B с С, what may we conclude?(4) How docs Ø stand in the relation?(5) How does N stands in the relation?3.1.1. Linear Orders. Certain types of relations have served important roles in analysis.Among them are orders, and we shall first study linear orders. A nondegenerate set is a setwith more than one element.Definition 3.3. Let X be a set and < a relation on X. We say < is a linear order iff itsatisfies the following axioms:(1) Antireflexive: Væ E X, ¬(x < x).(2) Comparable: V different x, y E X, x < y V y < x.(3) Transitive: V different x, y, z E X, (x < y ^y < z) = x < z.A set with a linear order is a pair (X,<). When we say X is a linearly ordered set, weunderstand a particular linear order < on X. When we say X is an ordered set we usuallymean X is a linearly ordered set.Remark 3.4. Given a set X and linear order <, for convenience we write a < b as shorthandfor a < b V a = b. The reader should be able to figure out the standard meaning of > and2.Remark 3.5. The natural order on the natural numbers N, conveniently symbolized by<, is a linear order. The natural order on the real numbers R is also a linear order. Theseexamples motivate the definition above.Exercise 3.6. Prove that a linear order is antisymmetric: V different x, y E X,¬(x <улу<х).Problem 3.7. Why is C not a linear order on the collection of subsets of N? Whichахiоms fail? ‑

Anti reflexive is the property which tells that nothing satisfies relation with itself. Same as…

Answered: Problem 3.7. Why is C not a linear…

Problem 3.7. Why is C not a linear order on the collection of subsets of N? Which axioms fail? O

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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