Is cardinality of sigma-algebra for two independent random variables is the sum of cardinalities of two sigma-algebras?

Let’s say that there are two independent r.v. $A$ and $B$. Is $card {sigma(A,B)}$ the sum of $card {sigma(A)}$ and $card {sigma(B)}$, where $card$ is cardinality (the number of elements in $sigma$)? Can you give a counter example or some ideas for the proof?