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 $\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \rbrace$.

(a) What is the multiplicative identity of $P^X$?

A. $\lbrace 1, 4, 8 \rbrace$

B. $\lbrace 1, 2, 3, 7, 10 \rbrace$

C. $X$

D. $\emptyset$

E. None of the above

(b) What is the additive identity of $P^X$?

A. $\lbrace 1, 2, 3, 4, 7, 9, 10 \rbrace$

B. $\emptyset$

C. $X$

D. $\lbrace 1, 4, 5 \rbrace$

E. None of the above

(c) Which of these are nonzero* zero divisors of $P^X$?

A. $\lbrace 5, 6 \rbrace$

B. $\lbrace 1, 3, 4, 6, 7, 8, 9, 10 \rbrace$

C. $\emptyset$

D. $\lbrace 1, 2, 6, 9 \rbrace$

E. $\lbrace 3, 4, 6, 7, 8, 9, 10 \rbrace$

F. $\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \rbrace$

G. None of the above

(d) Which of these are units of $P^X$?

A. $\lbrace 2, 4, 5, 6, 7, 8, 10 \rbrace$

B. $\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9 \rbrace$

C. $\lbrace 1, 2, 4, 5, 6, 8, 10 \rbrace$

D. $\lbrace 5, 7 \rbrace$

E. $\emptyset$

F. $X$

G. None of the above

*(Some authors define zero to be a zero divisor, others disagree.)