Given $P(A) = 0.9$ and $P(B) = 0.9$, what is a lower bound of $P(A \cup B)$? if $P(A) = 0.9$ and $P(B) = 0.9$ what is the lower bound for $P(A \cup B)$?

Comment: I agree with @soakley that you should have shown what you have tried. It seems you need to get more familiar with
the fundamental rules of probability. Such as:
$P(A \cup B) = P(A)+P(B) -P(A\cap B).$
If events $A$ and $B$ are the same, then $P(A\cap B) = P(A) = P(B) = 0.9,$ the smallest possible probability of $P(A\cap B),$ so $P(A\cup B) \ge 0.9.$
Maybe draw a Venn Diagram with events $A$ and $B$ and
imagine the events 'merging', so that the overlap region gets larger and larger--as the union gets ever smaller.
[But be sure to see both of @whuber's Comments.]
