2021/082) If,for all sets A, $A \cup B = A $ then prove that $B = \emptyset$

This is true.

To prove the same we have $A \cup B = A $ iff $B \subseteq A$

Let us take 2 sets $A_1,A_2$ which are disjoint and because it is true for every set $A_1 \cup B = A_1 $ so $B \subseteq A_1$

and $A_2 \cup B = A_2 $ so $B \subseteq A_2$

So from above 2 we have

$B \subseteq A_1 \cap A_2$

Because $A_1,A_2$ are disjoint sets so we have $A_1 \cap A_2= \emptyset$

So $B = \emptyset$