Method of verifying answers

Question

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 ?

Answer 1

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 $(\lnot A) \land(\lnot B)=\lnot(A\lor B)$

Answer 2

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 ....

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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.