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Exercise 3.6. Prove that a linear order is antisymmetric: V different x, y E X,-(x < y Ay < x).

Exercise 3.6. Prove that a linear order is antisymmetric: V different x, y E X,-(x < y Ay < x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Solve 3.6 and 3.8 in detail please 

3.1. Linear Order. Our goal is to define a topology on a set induced by an order on that set. 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 X. For x, y E X, rather than write (x, y) E R, we usually write xRy, putting the symbol of the relation between x and y. Exercise 3.2. Consider the relation "is a subset of" symbolized C on the collection of all subsets of N. Rather than write ({1,2}, {1, 2, 3}) eC, we write {1,2} C {1,2, 3}. For this relation, 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 B C A? (3) Given A, В,С subsets of N, if A c В аnd B c C, what may we coпcude? (4) How does 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 set with more than one element. Definition 3.3. Let X be a set and < a relation on X. We say < is a linear order iff it satisfies the following axioms: (1) Antireflexive: Vx 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, we understand a particular linear order < on X. When we say X is an ordered set we usually mean X is a linearly ordered set. Remark 3.4. Given a set X and linear order <, for convenience we write a < b as shorthand for a < b V a = b. The reader should be able to figure out the standard meaning of > and >. 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. These examples motivate the definition above. Exercise 3.6. Prove that a linear order is antisymmetric: V different x, y E X,¬(x < y ^y < x). Problem 3.7. Why is C not a linear order on the collection of subsets of N? Which ахiотs fail? ‑ Exercise 3.8. Let X be a set with linear order <. Let Y C X. Show that < induces a linear order on Y. (Define it compare "restricting" a function to a subset of its domain.)
Transcribed Image Text:3.1. Linear Order. Our goal is to define a topology on a set induced by an order on that set. 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 X. For x, y E X, rather than write (x, y) E R, we usually write xRy, putting the symbol of the relation between x and y. Exercise 3.2. Consider the relation "is a subset of" symbolized C on the collection of all subsets of N. Rather than write ({1,2}, {1, 2, 3}) eC, we write {1,2} C {1,2, 3}. For this relation, 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 B C A? (3) Given A, В,С subsets of N, if A c В аnd B c C, what may we coпcude? (4) How does 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 set with more than one element. Definition 3.3. Let X be a set and < a relation on X. We say < is a linear order iff it satisfies the following axioms: (1) Antireflexive: Vx 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, we understand a particular linear order < on X. When we say X is an ordered set we usually mean X is a linearly ordered set. Remark 3.4. Given a set X and linear order <, for convenience we write a < b as shorthand for a < b V a = b. The reader should be able to figure out the standard meaning of > and >. 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. These examples motivate the definition above. Exercise 3.6. Prove that a linear order is antisymmetric: V different x, y E X,¬(x < y ^y < x). Problem 3.7. Why is C not a linear order on the collection of subsets of N? Which ахiотs fail? ‑ Exercise 3.8. Let X be a set with linear order <. Let Y C X. Show that < induces a linear order on Y. (Define it compare "restricting" a function to a subset of its domain.)
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