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 X.For x, y E X, rather than write (x, y) E R, we usually write xRy, putting the symbol of therelation between x and y.Exercise 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}) eC, 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 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 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: 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, 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 > 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. Theseexamples 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? Whichахiотs fail? ‑Exercise 3.8. Let X be a set with linear order <. Let Y C X. Show that < induces alinear order on Y. (Define itcompare "restricting" a function to a subset of its domain.)

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Description: 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 se

Definition: Let X be a set and < a relation on X. We say < is a linear order if it satisfies…

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

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

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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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Question

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 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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