Confused by Metric Space Notation: What Does It Mean?

[rethipher]

I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying.

In regards to a Metric space

A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function

${{\bf{d: M \times M}}}$ (the syntax here)

such that for any x,y,z $\in \bf{M}$ (and the e looking symbol)

Does the times symbol in the first part indicate an M by M matrix, how would this be read. And the e symbol which I don't know the name of, I believe is akin to saying all numbers in, or every set in the space, or something roughly like that. So, how would I read the whole thing given the mathematical statement as written above if there was no explanation surrounding it? I have a lot of trouble reading pure math books sometimes because I don't understand a lot of the simple notation, even though I've seen the material before. I don't necessarily know how to represent the statement mathematically, for instance if I wanted to say something like a function that takes all real numbers as arguments, or considering all real numbers.

 

[pwsnafu]

I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying.

First thing I want to say, is congrats. Notation is something a lot of students underestimate, so I applaud you on this.

In regards to a Metric space

A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function

${{\bf{d: M \times M}}}$ (the syntax here)

such that for any x,y,z $\in \bf{M}$ (and the e looking symbol)

Does the times symbol in the first part indicate an M by M matrix, how would this be read. And the e symbol which I don't know the name of, I believe is akin to saying all numbers in, or every set in the space, or something roughly like that. So, how would I read the whole thing given the mathematical statement as written above if there was no explanation surrounding it? I have a lot of trouble reading pure math books sometimes because I don't understand a lot of the simple notation, even though I've seen the material before. I don't necessarily know how to represent the statement mathematically, for instance if I wanted to say something like a function that takes all real numbers as arguments, or considering all real numbers.

The times symbol, when used with sets, is called the Cartesian product. If you don't know set theory, then treat it as "d is a function of two variables, both chosen from M"

The e symbol is the "element of" symbol. $x \in X$ means that X is a set, and x is a member (element) of that set. For example, if X is the set denoting the integers, then x is an integer.

 

[micromass]

My recommendation to you is to first read a book on basic set theory before dealing with metric spaces. A book like "How to prove it" by Velleman should suit your purposes (although it contains much more material than is necessary).

 

[rethipher]

Thank you for the clear explanation, it is definitely helpful. My concern with the importance of notation stems more from me being easily confused by ambiguous notation and unclear notation than anything else.

Also, I looked up the set theory book, and it wasn't very expensive to get a used older version of it from 1994, so i went ahead and bought it. Thank you! I was having trouble finding set theory material to study because it seems like there is a large amount of set theory out there, and since it is used in basically every branch of mathematics, it's everywhere.

 

[CellCoree]

As a scientist familiar with mathematical notation, I can understand your confusion about metric space notation. Let me break it down for you.

First, the ordered pair (M,d) represents a metric space, where M is a set and d is a metric on that set. This means that d is a function that takes two elements from M and returns a distance between them.

The notation {{\bf{d: M \times M}}} indicates that d is a function that takes two arguments, both of which are elements of the set M. The \times symbol represents the Cartesian product, which is a way of combining two sets to form a new set. So, in this case, it means that d takes two elements from M and returns a distance between them.

The e symbol that you mentioned is the Greek letter epsilon (ε) and it represents "for all" or "for every". So, when we say "for any x,y,z \in \bf{M}", it means that we are considering all possible combinations of x, y, and z from the set M.

To read the whole statement, you can say "A metric space is an ordered pair (M,d) where M is a set and d is a function that takes two elements from M and returns a distance between them, for all x, y, and z in M." This means that in a metric space, we can calculate the distance between any two elements from the set M using the function d.

I understand that mathematical notation can be confusing, but with practice and understanding of the basic symbols, it becomes easier to read and interpret. I would suggest referring to a textbook or online resources for further clarification on specific symbols and their meanings.