For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.

If $A, B$ are elements of $P^X$, define

$$A + B := ( A - B ) \cup ( B - A )$$
$$A \times B := A \cap B$$

FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.

For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6,7} \rbrace$.

(a) How many elements are there in $P^X$?


Now let $A,B$ be two subsets of $X$ defined as follows:
$$A = \lbrace {1,3,4,5} \rbrace$$
$$B = \lbrace {1,3,5,7} \rbrace$$
Enter the elements of each set below as a comma-separated list. Input N for the empty set.

(b) What is the additive inverse of the subset $A$?
$\lbrace$ $\rbrace$

(c) What is $A + B$?
$\lbrace$ $\rbrace$

(d) What is $A \times B$?
$\lbrace$ $\rbrace$