Suppose $\mathfrak{A}$ is a free Boolean algebra and $G$ a countable set of free generators of $\mathfrak{A}$.
What is the cardinality of $\mathfrak{A}$ if $G$ is countably infinite, but we only allow finite unions and intersections of elements of $G$?
What is the cardinality of $\mathfrak{A}$ $G$ is countably infinite, but if this time we allow countably infinite unions and intersections of elements of $G$?