Defining logical form of statement: Difference between "are both not" and "are not both" 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.
I don't feel the difference between both statements.
My solution is: 
a)¬(A ∧ B)
b)(¬A ∧ ¬B)
Am i right? If not, can you explain why my solution is wrong? If I am right, explanation would be appreciated.
 A: Your solutions are correct.
Alice and Bob are not both in the room can be read as It is not the case that Alice and Bob are both in the room. This is true if, for example, Alice isn't in the room but Bob is - because they aren't both in the room, just one.
Alice and Bob are both not in the room can be read as Alice is not in the room and Bob isn't, either - this is true if neither Alice nor Bob is in the room, but if Alice isn't in the room and Bob is the statement must be false.
A: You are correct.
"(a) Alice and Bob are not both in the room."
means that either none of Alice and Bob are in the room, OR Alice alone is in the room, OR Bob alone is in the room.
"(b) Alice and Bob are both not in the room."
means that none of Alice and Bob are in the room.
A: Your solution is correct.
The principle is: 'not' negates the word/phrase immediately after it.
A: 
(a) Alice and Bob are not both in the room.
¬(A ∧ B)

Correct. The 3 possibilities include:

*

*Alice is in, and Bob is out

*Alice is out, and Bob is in

*Alice is out, and Bob is out

The only possibility excluded is only: Alice is in, and Bob is in.

(b) Alice and Bob are both not in the room.
(¬A ∧ ¬B)

Correct. The only possibility: Alice is out, and Bob is out.
