We know that one of the fundamental rule of counting is if A can be done in $m$ ways and B can be done in $n$ ways, then total number of ways of doing both is $ m \cdot n$. In terms of sets and cardinality, $ | A \times B | = | A | \times | B | $
Is this true for all non mutually exclusive events, A and B, or does this require events to be independent as well?
How do we show $| A \times B |$ on a Venn diagram?
$A\times B$ cannot be shown on a Venn diagram that also contains $A,B$, because its elements are not elements of $A$ nor $B$. Instead, elements of $A\times B$ are ordered pairs:
$$A\times B=\{(x,y):x\in A, y\in B\}$$
The product rule cited applies when we choose one from $A$ and also one from $B$. For example, if there are three types of bread, and two types of jelly, then there are $6=3\times 2$ ways to build a sandwich from a slice of bread and some jelly.