(Idempotent laws) For any set A, prove that:
$\left(i\right)A\cup A=A$

$\left(ii\right)A\cap A=A$

For any sets A and B, prove that:
$\left(i\right)A\cup (A\cap B)=A$

$\left(ii\right)A\cap (A\cup B)=A$

(De Morgan's laws) For any two sets A and B, prove that:
$I.(A\cup B{)}^{\prime }=(A^{\prime }\cap B^{\prime })$

$II.(A\cap B{)}^{\prime }=(A^{\prime }\cup B^{\prime })$

(Commultative laws) For any two sets a and B, prove that:
$I.A\cup B=B\cup A$ [Commutative law for union of sets]

$II.A\cap B=B\cap A$ [Commutative law for intersection of sets]

(Distributive laws) For any three sets A, B, C prove that:

I. $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
[Distirbutive law of union over intersection]

II. $A\cap \left[\right(B\cup C\left)\right]=(A\cap B)\cup (A\cap C)$
[Distributive law of intersection over union]