Chapter 3.1, Problem 43E

Write out the elements of $P(A)$ for the set $A=\left\{a,b,c\right\}$, and construct an addition table for $P(A)$ using addition as defined in Exercise $42$. (Sec. $1.1,\#7c$)

Sec. $1.1,\#7c\gg$

42. For an arbitrary set $A$, the power set $P(A)$ was defined in Section $1.1$ by $P(A)=\left\{X|X\subseteq A\right\}$, and addition in $P(A)$ was defined by

$X+Y=(X\cup Y)-(X\cap Y)$

$=(X-Y)\cup (Y-X)$

Prove that $P(A)$ is a group with respect to this operation of addition.

If $A$ has $n$ distinct elements, state the order of $P(A)$.