My Abstract Algebra professor was going through an example in class today that I really don't understand. The problem is:
Let $B(X)$ be the set of all subsets of $X$. Consider two binary operations on $B(X):$ $Y\cap Z$ and $Y\cup Z$, for $Y$, $Z\in B(X)$. Can these two operations be the binary operations of some ring on $B(X)$?
Can someone explain how I would start to begin this example or how to determine if this is, in fact, a ring? I think it's not a ring because I cannot figure out what the inverse would be.
Here is a simple diagram of the Power set of $\{a,b,c\}$
To find the union of two sets, then we follow the latice upward until we find an node above our sets with edges that connect to the set.
To find the intersection, move downward.
Hopefully this will help to visualize the problem.
To prove that it is a ring, intersection and union are both associative and commutative.
The distributive property holds $A\cap (B\cup C) = (A\cap B)\cup (A\cap C)$
That there is an identity element for each operation. What are your identity elements for each operation?
And that there is an inverse associated with one operation.
Rings do not require multiplicative inverses.