Well Orders and Total Orders .... Searcoid Definition 1.3.10 ....

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In summary, Micheal Searcoid's book "Elements of Abstract Analysis" discusses the concept of well-ordered and totally ordered sets. He provides a definition for each, and explains how to prove that every well-ordered set is totally ordered. When discussing singleton sets, he notes that they are still totally ordered. Peter provides access to definitions of well-ordered and totally ordered sets, and explains how to compare them to a singleton set.
  • #1
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I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ...

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.3 Ordered Sets ...

I need some help in fully understanding some remarks by Searcoid following Definition 1.3.10 ...

Definition 1.3.10 and the remarks following read as follows:
View attachment 8427
View attachment 8428In Searcoid's remarks following Definition 1.3.10 we read the following ...

"... ... every well ordered set is totally ordered ... ... Can someone please help me to prove that every well ordered set is totally ordered ... ...Help will be appreciated ...

Peter

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It may help MHB memebers reading the above post to have access to Definition 1.3.4 ... so I am providing access to the same ... as follows ...
View attachment 8429
Hope that helps ...

Peter
 

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Hi Peter,

You just have to apply the definition to subsets of two elements.

More explicitly, if $S$ is well ordered and $\{x,y\}\subset S$ with $x\ne y$, then $\{x,y\}$ has a smallest element. If that element is $x$, then $x<y$; if that element is $y$, then $y<x$.
 
  • #3
castor28 said:
Hi Peter,

You just have to apply the definition to subsets of two elements.

More explicitly, if $S$ is well ordered and $\{x,y\}\subset S$ with $x\ne y$, then $\{x,y\}$ has a smallest element. If that element is $x$, then $x<y$; if that element is $y$, then $y<x$.

Sorry to butt in.. But what if $S$ is singleton? They also have a minimum value since no number is greater than itself (...). How do we compare it to a totally ordered set which (seems to) require at least two elements?

edit: i.e. if it doesn't make sense consider a totally ordered set which is singleton, how can we say all well ordered sets are totally ordered? (I'm assuming we can't and that I've missed a definition or am thinking something really silly)..
 
  • #4
Joppy said:
Sorry to butt in.. But what if $S$ is singleton? They also have a minimum value since no number is greater than itself (...). How do we compare it to a totally ordered set which (seems to) require at least two elements?

edit: i.e. if it doesn't make sense consider a totally ordered set which is singleton, how can we say all well ordered sets are totally ordered? (I'm assuming we can't and that I've missed a definition or am thinking something really silly)..
Hi Joppy,

A singleton set is totally ordered, because the definition is vacuously true.

The definition of a totally ordered set $S$ is equivalent to: if $\{x,y\}\subset S$ and $x\ne y$, then $x<y$ or $y<x$.

If $S$ is a singleton, the condition $x\ne y$ is always false. As the antecedent of the implication is false, the implication itself (the definition) is true.

Note that it is quite common to define an order relation as a reflexive, antisymmetric and transitive relation (like $\le$), and to modify the definitions accordingly. In that case, things are a little simpler.
 

FAQ: Well Orders and Total Orders .... Searcoid Definition 1.3.10 ....

1. What is a well order?

A well order is a type of mathematical ordering system in which every non-empty subset of a set has a least element. This means that every element in the set can be compared to others in a specific order.

2. How is a well order different from a total order?

While both well orders and total orders involve a specific ordering of elements, a total order requires that all elements in a set can be compared to each other, while a well order only requires that each subset has a least element.

3. What is the Searcoid Definition 1.3.10?

The Searcoid Definition 1.3.10 is a specific definition used in the study of well orders and total orders. It states that a well order is a total order in which every non-empty subset has a least element.

4. What are some real-world examples of well orders?

One example of a well order in the real world is the ranking system used in sports tournaments, where each team is compared to others in a specific order and there is a least-ranked team. Another example is the ordering of items in a menu at a restaurant, where each dish is listed in a specific order and there is a least expensive option.

5. How are well orders and total orders used in science?

In science, well orders and total orders are often used in the study of sets and their elements. These ordering systems can help scientists understand the structure and relationships between different elements in a set, and can also be used in mathematical proofs and calculations.

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