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Why isn't independence defined as $P(A \cap B) = P(A) ? P (B)$? 

In the title, ? = one of the four elementary arithmetic operations. Why isn't independence defined as $P(A \cap B) = P(A)$

1. $+ P (B)$?

2. or $- P (B)$?

3. or $÷ P (B)$?

>### Definition 2.5.1 (Independence of two events). 

>Events A and B are _independent_ if
$P(A \cap B) = P(A)P(B)$.

>If P(A) > 0 and P(B) > 0, then this is equivalent to
$P(A|B) = P(A)$;
and also equivalent to $P(B|A) = P(B)$.

Blitzstein, *Introduction to Probability* (2019 2 ed), p 63. 

>We have introduced the conditional probability P(A|B) to capture the partial
information that event B provides about event A. An interesting and important
special case arises when the occurrence of B provides no such information and
does not alter the probability that A has occurred, i.e. ,
P(A I B) = P(A).
When the above equality holds. we say that A is **independent** of B. Note that
by the definition $P(A | B) = P(A \cap B)/P(B)$, this is equivalent to
$P(A \cap B) = P(A)P (B).$
We adopt this latter relation as the definition of independence because it can be
used even when P(B) = 0, in which case P(A|B) is undefined. The symmetry
of this relation also implies that independence is a symmetric property; that is,
if A is independent of B, then B is independent of A, and we can unambiguously
say that A and B are **independent events**.

Tsitsiklis, *Introduction to Probability* (2008 2e), p 34.