separating propositons with commas? From Kaye's Mathematics Logic, about notation for propositional logic:

Another place where we relax notation is in the notation on the left hand
  side of a turnstile symbol $\vdash$. Instead of using set theory notation with $\{\ldots\}, \cup,
∅$, etc., it is traditional to list formulas and sets of formulas, separating them
  with commas, and regard the list as a single set of formulas, so the order of
  formulas in the list and any repetitions in it is ignored. This applies to both
  the turnstile $\vdash$ of this chapter and the turnstile that will be introduced in the
  next. Thus, with all the conventions in place, the previous example would be
  written as $a ∧ b \vdash ¬(¬ a ∨ ¬ b)$. The empty set is written as an empty list, as in $\vdash (a ∨ ¬ a)$.



*

*Does "separating them with commas" mean to represent $a ∧ b \vdash
    ¬(¬ a ∨ ¬ b)$ as $a, b \vdash ¬(¬ a ∨ ¬ b)$ ? Then what is the
difference between  $a ∧ b$  and $a, b$ ?
I can't figure out what the paragraph is saying from the two
examples.

*Also a propositional language doesn't have comma as a
punctuation symbol, while a first order language does. Is it true?


Thanks.
 A: The "official" notation is :

$\Gamma \vdash \varphi$

where :

$\Gamma = \{ φ_1,\ldots,φ_n \}$.

The "abbreviation" licenses us to write it as :


$φ_1,\ldots,φ_n \vdash φ$.


Also, when $\Gamma = \emptyset$, instead of :

$\emptyset \vdash \varphi$

we will write :

$\vdash \varphi$.

Another abbreviation is the following :

$\Gamma, \alpha \vdash \varphi$

in place of the "official" :

$\Gamma \cup \{ \alpha \} \vdash \varphi$.


Comment
Regarding the example discussed, it must be :

$\{ a∧b \} \vdash a$

abbreviated as :

$a∧b \vdash a$

and not : $a, b \vdash a$. 
We have to note an important distinction: that between object language and meta-language.
In the formal system of propositional logic, assuming that the propositional letters are the $p_i$'s, we have that e.g. $p_1 \land p_2$ is a formula.
The symbol :
$\Gamma \vdash \varphi$ 
is not part of the formal system. It is part of the meta-language, and says that there is a derivation (i.e.a sequence of formulae of the system satisfying certain rules) of the formula $\varphi$ from the assumptions in $\Gamma$.
Thus, when we license ourselves to "abbreviate" :

$\{p \rightarrow q, q \rightarrow r \} \vdash p \rightarrow r$

as :

$p \rightarrow q, q \rightarrow r \vdash p \rightarrow r$

we are introducing an abbreviation in the metalanguage.

Added
About commas in f-o logic : yes, we need it for terms, written as :

$f_i(x_1, \ldots, x_n)$

where $f_i$ is the $i$-th function letter (in the enumeration of the alphabet) and has $n$ argument-places.
If we choose to make explicit the number of arguments, writing : $f_i^n$, we can avoid comma and parentheses, writing terms as :

$f_i^nx_1 \ldots x_n$.

