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.
