Method of verifying answers I have been reading Velleman's How to prove it book and solving problems of the exercise in it.
What concerns me is that I cannot verify if actually my solutions are correct. The book has only solutions to selected problems.How do mathematicians verify that if their answer is correct ?
For example, one of their question is like this:

Analyze the logical forms of the following statements: (a) Alice and
  Bob are not both in the room. (b) Alice and Bob are both not in the
  room.

Now, I feel that both the statements are equivalent and have come up with this answer:

A = Alice is in the room. B = Bob is in the room.
(a) ¬(A ∧ B)
(b) ¬(A ∧ B)

But I'm not sure whether they are correct. And there may be lots of other problems like this. How to approach these situations generally ?
 A: The statements are not equivalent.
For the first statement you can see that not both, this is the opposite as they're both in the room which would be $A\land B$ and negating that gives $\lnot(A\land B)$
For the second statement however, both not means that it has to be true for both of them not being in the room, so it is $(\lnot A) \land(\lnot B)=\lnot(A\lor B)$
A: Consider: (a) Alice and Bob are not both in the room. (b) Alice and Bob are both not in the room.

I feel that both the statements are equivalent ...

Really? Really?? What do you "feel" about (a*) Alice and Bob are not both boys vs (b*) Alice and Bob are both not boys? Or about (a**) Alice and Bob are not both top of the class vs  (b**) Alice and Bob are both not top of the class? 
Think about it: if Alice is in the room and Bob outside, then (a) is true and (b) is false, so of course the claims are not equivalent ....

More generally, if you want to check your answers to exercises in Velleman, you will find a long series of blog posts starting here useful.
