Defining mathematical objects

[X89codered89X]

I was wondering what the typical approach is for creating a "mathematical object" such as the probability model kolmogorov made (I've also heard it called a probability space...not really sure what the difference is)...

$<\Omega,\mathcal{F},P>$, where $\Omega$ is the sample space, $\mathcal{F}$ is the $\sigma$-field over $\Omega$, and $P$ is the map $P : \mathcal{F} \rightarrow [0,1]$. The idea being that 2 probability models are the same only if each of these 3 things are all identical...

Can you make groups like this for anything? e.g...

...I read in my textbook that "Two functions are the same if and only if they have the same Domain, Codomain, and Rule mapping from the Domain to the Co-Domain." ...Does this mean that I could augment a given function defined unrigorously as y = f(x) as a similarly constructed object like, say...

$<{\textbf{X}},{\textbf{Y}},f>$, where ${\textbf{X}}$ is the domain, ${\textbf{Y}}$ is the co-domain, and $f$ maps from the domain to the co-domain .

I realize this is often a pointless model to make ( other than in probability which has complications in defining the domain )... you can say all this with just ... $f: {\textbf{X}} \rightarrow {\textbf{Y}}$ but aside from its unnecessary-ness. is there any reason I can't do this...

Where can I study this in more depth/ what's studying functions in this depth called? Any recommended Texts on this area?

Thank you All!

 

[Stephen Tashi]

The idea being that 2 probability models are the same only if each of these 3 things are all identical...

The word "same" is slang. What you are stating is technically a definition for an "equivalence relation" on probability models. When a given equivalence relation is understood then we can talk about two things being "equal" with respect to that equivalence relation. For example, 2+2 is equal to 4 using the usual equivalence relation defined on real numbers, but "2+2" is not equal to "4" using the usual equivalence relation defined on strings of characters.


If you want to study the logical technicalities of math, you should study a little mathematical logic. Then study a book that treats some area of mathematics (such as set theory or abstract algebra) in a rigorous manner. I don't know what selection of modern books is available. Perhaps other forum members have suggestions.

Math books that are fairly rigorous often don't treat elementary concepts rigorously. They assume you can handle them informally. You can define a function as a triple of things and use the triple to define an equivalence relation on functions. I've not often seen it done this way.

I've never seen a formal definition of a "mathematical object". It's true that many things in mathematics can be defined as tuples that consist of other things.

 

[micromass]

I was wondering what the typical approach is for creating a "mathematical object" such as the probability model kolmogorov made (I've also heard it called a probability space...not really sure what the difference is)...

$<\Omega,\mathcal{F},P>$, where $\Omega$ is the sample space, $\mathcal{F}$ is the $\sigma$-field over $\Omega$, and $P$ is the map $P : \mathcal{F} \rightarrow [0,1]$. The idea being that 2 probability models are the same only if each of these 3 things are all identical...

Can you make groups like this for anything? e.g...

...I read in my textbook that "Two functions are the same if and only if they have the same Domain, Codomain, and Rule mapping from the Domain to the Co-Domain." ...Does this mean that I could augment a given function defined unrigorously as y = f(x) as a similarly constructed object like, say...

$<{\textbf{X}},{\textbf{Y}},f>$, where ${\textbf{X}}$ is the domain, ${\textbf{Y}}$ is the co-domain, and $f$ maps from the domain to the co-domain .

I realize this is often a pointless model to make ( other than in probability which has complications in defining the domain )... you can say all this with just ... $f: {\textbf{X}} \rightarrow {\textbf{Y}}$ but aside from its unnecessary-ness. is there any reason I can't do this...

Where can I study this in more depth/ what's studying functions in this depth called? Any recommended Texts on this area?

Thank you All!

Books on set theory will go more in depth to defining elementary notions such as functions and relations. I can highly recommend "Introduction to Set Theory" by Hrbacek and Jech.