# Ohda: Definition of Order in Baby Rudin

• Bacle2
In summary, Rudin defines order as an order relation < in a set S, satisfying certain properties, and only considers total orders in his "Baby Rudin" book, possibly in order to stay close to the intuition of the order relation on the reals. He may use the subset relation in his "Non-Baby" book for proofs such as Hahn-Banach.
Bacle2
Hi, All:

Just curious:

Rudin defines order in his "Baby Rudin" book ; an order relation < in a set S, as a relation* satisfying, for any x,y,z on S:

1) Either x<y , y<x , or y=x

2)If x<y and y<z , then x<z , i.e., transitivity.

Just curious: why is Rudin only considering only total orders in his book? Isn't the partial-order relation of "is a subset of" (among others) important-enough to allow for partial orders?

* Rudin never formally-defined relation, just in case, tho let's assume a relation

on S is a subset of SxS with the above properties.

I suppose that it is sufficient for his goals. Certainly the subset relation is very interesting, but he probably doesn't need it anywhere in his book.

I guess he wants to stay close to the intuition of the order relation on the reals...

You may be right. Still, AFAIK, you need to work with set containment to prove, e.g., Hahn-Banach. Maybe he does that in his "Non-Baby" book.

## 1. What is Ohda in Baby Rudin?

Ohda refers to the concept of order in Baby Rudin, a textbook on mathematical analysis written by mathematician Walter Rudin. It is used to describe the arrangement or sequence of elements in a set or collection.

## 2. How is Ohda defined in Baby Rudin?

In Baby Rudin, Ohda is defined as a relation on a set that is reflexive, transitive, and antisymmetric. This means that it relates each element in the set to itself, and if one element is related to another, then the second element is also related to the first. Additionally, if two elements are related to each other, they must be the same element.

## 3. Why is Ohda important in mathematics?

Ohda is important in mathematics because it allows us to study and understand the order and structure of sets. It is a fundamental concept in many branches of mathematics, including analysis, algebra, and topology. Ohda also plays a crucial role in the development of mathematical theories and proofs.

## 4. How is Ohda used in real-world applications?

Ohda has many real-world applications, especially in computer science and data analysis. It is used to sort and organize data in databases and algorithms, making it easier to search and retrieve information. Ohda is also used in economics and game theory to model and understand decision-making processes.

## 5. Are there any variations of Ohda in mathematics?

Yes, there are variations of Ohda in mathematics, including strict and partial orders. Strict orders do not allow for an element to be related to itself, while partial orders allow for incomparable elements. These variations have their own properties and applications in different areas of mathematics.

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