Is Every Totally Ordered Set Well-ordered?

by | Last updated on January 24, 2024

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By definition,

any well-ordered set is totally ordered

. However, the converse is not true – the set of integers which is totally ordered, is not well-ordered under the standard ordering (since itself and some its subsets do not have least elements). Although, any finite totally ordered set is well-ordered.

Which sets are well-ordered?

In general, a

set (such as N) with some order (<)

is called well-ordered if any nonempty subset has a least element. The set of even numbers and the set {1,5,17,12} with our usual order on numbers are two more examples of well-ordered sets and you can check this.

How do you determine if the set is well-ordered?

A set of real numbers is said to be well-ordered

if every nonempty subset in it has a smallest element

. A well-ordered set must be nonempty and have a smallest element. Having a smallest element does not guarantee that a set of real numbers is well-ordered.

Are ordinals well-ordered?

Definition. An ordinal is a transitive set that

is well-ordered by ∈

. (Frequently we will write < instead of ∈ when we are dealing with ordinals.)

What is not a well-ordered set?

Every finite totally ordered set is well ordered.

The set of integers

. , which has no least element, is an example of a set that is not well ordered. An ordinal number is the order type of a well ordered set.

Are integers well-ordered?

The well-ordering principle says that

the positive integers are well-ordered

. An ordered set is said to be well-ordered if each and every nonempty subset has a smallest or least element. … The set of positive integers does not contain any infinite strictly decreasing sequences.

Is every well-ordered set well founded?

The well-ordering theorem, which is equivalent to the axiom of choice, states

that every set can be well ordered

. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

What is meant by well-ordered?

Definition of well-ordered

1 :

having an orderly procedure or arrangement

a well-ordered household. 2 : partially ordered with every subset containing a first element and exactly one of the relationships “greater than,” “less than,” or “equal to” holding for any given pair of elements.

Is Z well-ordered set?

The set of integers Z

is not well-ordered under

the usual ordering ≤.

Can rational numbers be well-ordered?

The rationals, for example, do not form a well-ordering under the usual less-than relation, but there is a way of putting them into one-to-one correspondence with the natural numbers, so it

can be well-ordered by the total order implied

by this correspondence.

Is 0 1 A well ordered set?

We can say that the set of real numbers [0,1

] is not a well ordered set

as (0,1) is a subset of [0,1] and doesn’t have a least element but if this is only taken for integers, then it is well ordered set. but if we take (0,1) for integers , it is well ordered .

Is Empty set well ordered?

Note that

every well ordered set is totally ordered

, and that if X is empty, then the unique (empty) ordering on X is a well ordering.

What is meant by well-ordering list few examples?


A set of numbers is well ordered when each of its nonempty subsets has a minimum element

. The Well Ordering Principle says that the set of nonnegative integers is well ordered, but so are lots of other sets. For example, the set of numbers of the form , where is a positive real number and n ∈ N .

What is well-ordering property of N?

The well-ordering property of N states that “

For all sets S ⊆ N such that S = ∅, there exists a least element m ∈ S such that m ≤ t for all t ∈ S

.”

Is well-ordering principle an axiom?

This principle can be taken as an axiom on integers and it will be the key to proving many theorems. As a result, we see that

any set of positive integers is well ordered

while the set of all integers is not well ordered. If s objects are placed in k boxes for s>k, then at least one box contains more than one object.

Was well-founded?

If you say that a report, opinion, or feeling is well-founded, you mean that

it is based on facts and can therefore be justified

.

Is well-founded hyphenated?

Tips: When something is well-founded, it is based on facts or solid evidence.

Used after a verb, it is not hyphenated

.

What do you call to a set of two well ordered real numbers?

When we speak of the

Cartesian Coordinate Plane

, we mean the set of all possible ordered pairs (x,y) as x and y take values from the real numbers. Below is a summary of important facts about Cartesian coordinates.

Why are rational numbers not well ordered?

While many subsets of Z has a smallest element, the set Z itself does not have a smallest element. The rationals Q are not well-ordered:

The set Q itself does not have a smallest element

.

Are non negative integers well ordered?

The well ordering principle is that

every nonempty set of nonnegative integers has a least element

.

What is a simply ordered set?

A simply ordered set M is

a set such that if any two of

.

its elements are given it is known which one precedes

. A subset of M is said to be cofinal (coinitial) with M if no element of M follows (precedes) all the elements of the subset.

Juan Martinez
Author
Juan Martinez
Juan Martinez is a journalism professor and experienced writer. With a passion for communication and education, Juan has taught students from all over the world. He is an expert in language and writing, and has written for various blogs and magazines.