Given two sets A and $B,$ define $AB$ as follows:

$AB=\{x$:$(xinA$ and $x!$inB$)or(xinB$ and $x!$inA$)\}.$

$($a$)$ For generic sets A and $B,$ draw and shade the Venn diagrams corresponding to $AB,$

$AA,$ and $A\frac{O}{?}.$ Hypothesize general statements about what $AA$ and $A\frac{O}{?}$ equal,

then prove your result.

$($b$)$ For any sets A and $B,$ show that

$AB=(\frac{A}{\frac{?}{?}}B)(\frac{B}{\frac{?}{?}}A)$

$($c$)$ Show that $(AB)(BC)=AC$ for any sets $A,B,$ and $C.$ For this problem you may

assume, without proof, that $$ is associative; namely that $A(BC)=(AB)C.$

$($a$)$ Consider a compound predicate $S$ consisting of two constituent predicates $P$ and $Q.$ The

predicate $S$ returns true precisely when an even number of its constituent predicates are

true. Using only the classical operations of AND $({?}^{{?}^{?}}),$ OR $($ V $),$ and NOT $(not),$ construct

$S.$ Use a truth$-$table to verify that your answer is correct.

$($b$)$ Now consider the case of a compound predicate $U$ which consists of three constituent

predicates $P,Q,$ and $R,$ and returns true precisely when an even number of its con$-$

stituent predicates are true. Let $S$ be the compound predicate you constructed in part

$($a$).$ Using only the predicates $R$ and $S,$ and the classical operations of AND $({?}^{{?}^{?}}),$ OR

$($v$)$ and NOT $(not),$ construct $U.$ Use a truth$-$table to verify that your answer is correct.