집합의 연산

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$A - \left(A \cap B^c \right) = A \cap B$

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A^c \cup B \right)^c \cap A = A - B$

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

$\left( A \cup B \right) - A = B - A$

Let $A$ and $B$ be subsets of some universal set $U$. Prove or disprove each of the following:

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

Prove the following propositions:

If $A$, $B$, $C$, and $D$ are sets with $A \approx B$ and $C \approx D$ and if $A$ and $C$ are disjoint and $B$ and $D$ are disjoint, then $A \cup C \approx B \cup D$.