Show that the following four conditions are equivalent:

$(i)A \subset B\,\,\,({\rm{ ii }})A – B = \phi \quad (iii)A \cup B = B\quad (iv)A \cap B = A$

Find the union of each of the following pairs of sets :

$X =\{1,3,5\} \quad Y =\{1,2,3\}$

If $A = \{ x:x$ is a natural number $\} ,B = \{ x:x$ is an even natural number $\} $ $C = \{ x:x$ is an odd natural number $\} $ and $D = \{ x:x$ is a prime number $\} ,$ find $B \cap C$

If $A = \{2, 3, 4, 8, 10\}, B = \{3, 4, 5, 10, 12\}, C = \{4, 5, 6, 12, 14\}$ then $(A \cap B) \cup (A \cap C)$ is equal to

For any sets $\mathrm{A}$ and $\mathrm{B}$, show that

$P(A \cap B)=P(A) \cap P(B).$