(a,b)* is actually you have written in “easier to write for self understanding“ notation type way of representing “reg Language “ in which comma is understood as union ,although it is generally not seen in ToC and programming . A “union” can be represented using square brackets “[]” like [a-z] or using
$\cup$ or using “+” like (a+b) , according to the use and the convention.
in regex, whereas “,” is just a comma
[a-z] a very famous regex (programming) , means at-least 1 literal from “a or b or or ….z” ,which is quite similar to (a+b+...+z) in ToC
A reg language can be represented in Grammar form or DFA or NFA or e-NFA or in Regex .
Now in terms of ToC,
(a+b)* means any number of times either a or b or nothing : $\epsilon$ , a , aa… , aaa...b , b , bb , bbb …… , bbb…..a , ab ,ba , aa..b… , b...aa…. , likewise
(a*b*)* means any number of times a , followed by any number of times b : a , aa… , aaa...b , b , bb , bbb …… , bbb…..a , ab , but not ba or b...a…
{a,b}* , means language(or universal language) over the set of alphabets {a,b} , ={ $\epsilon$ , a , aa… , aaa...b , b , bb , bbb …… , bbb…..a , ab ,ba , aa..b… , b...aa…. } = $\Sigma^1$ $\cup$ $\Sigma^2$ $\cup$ $\Sigma^3$ $\cup$... $\Sigma^5$ $\cup$ ….
Check the brackets and the operators carefully.
(a+b)* $\neq$ (a*b*) $\neq$ {a,b}*
but ,
(a*b*) $\subset$ (a+b)* $\subset$ {a,b}*
Also , (a*b*) $\neq$ (a*b*)* , but (a*b*) $\subset$ (a*b*)*
also , (a+b)* = (a*b*)*
