# Ring structure on power set of a set.

Let $$A$$ be a non empty set. Let $$P(A)$$ denote the power set of $$A$$.

$$P(A)$$ can be given a group structure in multiple ways.

1. Using Disjoint union a group operation Source

may be worth noting in this context that the power set of $$X$$ is a group under the operation of disjoint union, with the empty set as identity. if the complement of $$A \subset X$$ is $$A'$$ then the disjoint union of $$A$$ and $$B$$ is: $$A \circ B = (A \cap B') \cup (A' \cap B)$$ proving associativity is a nice exercise, though it follows much more simply if the disjoint union is viewed as ordinary addition (mod 2) in $$\mathbb{F}_2^X$$

2. Using Symmetric Difference as Group Operation Source

The power set $$G$$ of any set $$A$$ becomes an abelian group under the operation of symmetric difference:

• Why abelian? Easy to justify, just use the definition in $$(1)$$ above: it's defined in a way that $$g_1 \triangle g_2$$ means exactly the same set as $$g_2 \triangle g_1$$, for any two $$g \in G$$.

• As you note, the symmetric difference on $$G$$ is associative, which can be shown using the definition in $$(1)$$, by showing for any $$f, g, > h \in G, (f\; \triangle\; g) \triangle \;h = f\;\triangle\; (g > \;\triangle\; h)$$.

• The empty set is the identity of the group (it would be good to justify this this, too), and

• every element in this group is its own inverse. (Can you justify this, as well? Just show for any $$g_i \in G, g_i\;\triangle \; g_i = > \varnothing$$).

Question: Can we define a ring structure on power set of $$A$$.