Simple probability question that's really annoying me

I'm not a stats type of guy so this problem is really getting under my skin. Please can someone explain this to me.

$P(A)= 0.65$ and $P(B)= 0.45$ then $P(A \&B )$ can have the smallest and largest values of 0.05 and 0.45 respectively.

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The smallest value is 0.10. –  Did Oct 25 '12 at 11:34

Suppose that the event $B$ is a subset of the event $A$: whenever $B$ happens, then $A$ also happens (but not necessarily vice versa). Then the event $A\&B$ occurs precisely when $B$ occurs: if $B$ occurs, so does $A$, and therefore so does $A\&B$, and if $B$ does not occur, then neither does $A\&B$. Clearly, then, $P(A\&B)=P(B)=0.45$ in this case. We’ve maximized the chance of getting both $A$ and $B$ by giving the two events the maximum possible overlap.
Now suppose that we go to the other extreme and give $A$ and $B$ the minimum possible overlap. Since $P(A)+P(B)=0.65+0.45=1.10$ is greater than $1$, they have to have some overlap. In fact, that extra $0.10$ must be overlap: the probability of $\text{not-}A$ is $1-0.65=0.35$, so even if $B$ includes every outcome that isn’t in $A$, it still has to include some outcomes in $A$ to bring its probability up to $0.45$. Specifically, the overlap has to have probability $0.10$, since it must make up the difference between $P(B)=0.45$ and $P(\text{not-}A)=0.35$.