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

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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 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.
For x, y E X, rather than write (x, y) E
relation between x and y.
we usually write xRy, putting the symbol of the
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}) €C, 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 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 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: 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, 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
2.
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 <
улу<х).
Problem 3.7. Why is C not a linear order on the collection of subsets of N? Which
ахiоms fail? ‑
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. For x, y E X, rather than write (x, y) E relation between x and y. we usually write xRy, putting the symbol of the 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}) €C, 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 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 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: 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, 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 2. 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 < улу<х). Problem 3.7. Why is C not a linear order on the collection of subsets of N? Which ахiоms fail? ‑
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