Difference between P(A,B) vs P(A∩C) vs P(A.B) vs P(AB)

Question

• P(A,B)
• P(A∩B)
• P(A.B)
• P(AB)

Above 4 statements looks almost similar to me. Can anybody define if there is any difference between above 4 and compare them in detail.

Answer 1

$A \cap B$ is often abbreviated to $AB.\;$ I also, am unfamiliar with $A.B$, but I suspect you may mean $A\cdot B$, which would be equivalent to the other two.

Then comma notation is very common with events involving random variables. For example $P(X \le 3, Y > 2)$ is common shorthand for $P(\{X \le 3\}\cap \{Y > 2\}).$ I have not seen it used just for letters as in $P(A,B),\,$ except for consistency in a discussion of random variables.

In summary, I think they are all the same, with the possible exception of the one with the dot in period position.

Notes:

(a) In older books, you may find $A + B$ as an abbreviation for $A \cup B$, and more recently, as an abbreviation for "$A \cup B$ where $A$ and $B$ are mutually exclusive". So that $P(A + B) = P(A) + P(B)$ as in Kolmogorov's axiom, without elaboration. But $P(A \cup B) = P(A) + P(B) - P(AB)$ is a theorem. Whenever you see $+$ between sets, you need to check the context very carefully.

(b) The use of $AB$ for $A \cap B$ can avoid parentheses in expressions such as $AB \cup CD = (A\cap B) \cup (C\cap D),$ because one takes it for granted that $\cap$ written as a 'product' precedes $\cup$ in the order of set-theoretic operations.