Answer
a) A - B = { x | (x $\in$ A) $\wedge$ (x $\notin$B)} = {x | x$\in$ A $\wedge$ x $\in \bar{B}$} = A $\cup \bar{B}$
b) $ (A \cap B) \cup (A\cap \bar{B})$
= $((A\cap B) \cup A) \cap ((A\cap B)\cup \bar{B})$
= $(A \cup (A\cap B) ) \cap ((A\cap B)\cup \bar{B})$
= $(A ) \cap ((A\cap B)\cup \bar{B})$
= $(A ) \cap (\bar{B} \cup (A\cap B))$
= $(A ) \cap (( \bar{B} \cup A)\cap (\bar{B} \cup B))$
= $(A ) \cap (( \bar{B} \cup A)\cap (U))$
= $(A ) \cap ( \bar{B} \cup A)$
= $(A ) \cap (A \cup \bar{B})$
= A
Work Step by Step
a) A - B means element that are in A but not in B. If an element is not in B, it is in $\bar{B}$.
b) To prove this, we will start on the left hand-side and reach the right-hand side.
Step 1 = Distributive law
Step 2 = Commutative law
Step 3 = Absorption law
Step 4 = Commutative law
Step 5 = Distributive law
Step 6 = Complement law
Step 7 = Identity law
Step 8 = Commutative law
Step 9 = Absorption law
