Why Do We Have Different Terms for Least Upper Bound and Supremum?

[back2square1]

Are the least upper bound and supremum of a ordered field same thing? If so, then why do we have two different terms and why do textbooks do not use them interchangeably. That also means that greatest lower bound and infimum are also the same thing.

 

[skiller]

Just a couple of expressions to say the same thing.

 

[HallsofIvy]

As far as I know "supremum" and "least upper bound" are terms for the same thing. Similarly "infimum" and "greatest lower bound" are terms for the same thing.

Surely you don't think it is unusual to have two different terms for the same thing? And I have never seen a text that did NOT "use them interchangebly" where it used both. Commonly, 'lower level' texts use the English phrase "greatest lower bound" and 'upper level' texts use the latin word "supremum" (which is also commonly used internationally).

 

[back2square1]

Thanks you so much. My textbook uses both but they are very specific about where they use which term. And, I'm a beginner in Mathematical Analysis.

 

[blue_raver22]

The terms "least upper bound" and "supremum" are used interchangeably in mathematics and refer to the same concept. They both represent the smallest element in a set that is greater than or equal to all other elements in that set. Similarly, "greatest lower bound" and "infimum" refer to the largest element in a set that is less than or equal to all other elements in that set.

The reason why these terms are used interchangeably is because they are both used to define the same concept, and therefore can be used interchangeably without any loss of meaning. However, it is important to note that some textbooks or mathematical contexts may prefer one term over the other, but this does not change the underlying concept.

The use of these different terms may also be due to historical reasons or personal preference of the author. Ultimately, it is important to understand the concept behind these terms rather than get caught up in the terminology.

In summary, the least upper bound and supremum, as well as the greatest lower bound and infimum, refer to the same concept in an ordered field. While some textbooks or contexts may use one term over the other, they can be used interchangeably without any loss of meaning.