I'm studying for an abstract algebra exam and one of the review questions was this:
Let $X$ be a set, and $\mathcal P(X)$ be the power set of $X$. Consider the operations $\Delta$ = symmetric difference (a.k.a. "XOR"), and $\bigcap$ = intersection.
a) Does $\Delta$ and $\bigcap$ make $P(X)$ into a ring?
b) If so, is it a ring with unity?
c) Is the ring commutative?
d) Is it a field?
For parts a) and b), I think it does form a ring with unity, but I'm not quite sure how to get started on proving it.
For part c), it is a commutative ring since $\mathcal P(X)$ is closed under symmetric difference and intersection, right?
Not even sure how to get started on d).
I'd really like to understand this question fully, so any kind of input would be tremendously helpful. Thank you!