For this question I'm experiencing a lot of difficulty. Let me preface this question. The previous part of the question stated that "R is a family of subsets of X (Which makes it a Ring) if it is closed under finite unions and differences." We then had to prove that R is closed under finite intersections.
I didn't really have a hard time proving that part of the question. The part that I am getting stuck on is proving that the ring R is an algebra if and only if X $\in$ R. Sorry if this is obvious and trivial but I just can't seem to get it.
By definition, if $R$ is an algebra of subsets of $X$ then $X\in R$. Conversely, if $R$ is a ring and $X\in R$ then, since $R$ is closed under differences, $A^c=X\setminus A\in R$ for all $A\in R$. It follows that, if $A,B\in R$, then $A\cap B=(A^c\cup B^c)^c\in R$. Hence $R$ is an algebra.