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Let $A$= Alice is in the room and $B$ = Bob is in the room

a. Alice and Bob are not both in the room

b. Alice and Bob are both not in the room

c.Either Alice or Bob is not in the room

d.Neither Alice nor Bob is in the room

My attempt :

a. $\neg$ (A $\land$ B)

b. $\neg$ A $\land$ $\neg$ B

c. $\neg$ A $\lor$ $\neg$ B

d. same as b

Thanks for help

J. Deff
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  • All looks good to me. You can check yourself by considering when each of your answers is true and confirming that that matches the question statement (e.g. for a, your answer is true exactly when A is false and B is false, i.e. when Alice is not in the room and Bob is not in the room, which is what was required) – postmortes May 24 '17 at 08:32
  • "Either Alice or Bob is not in the room" could be restated as "Alice is present and Bob is not; or, Alice is missing and Bob is present". Alternatively, "Alice is present, or Bob is present; and, they are not both present". – DanielV May 24 '17 at 11:46

1 Answers1

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Point c. is discussable

c. Either Alice or Bob is not in the room. This could actually additionally mean that it is not allowed that both are not in the room. This depends on if you interpret either or as exclusive or denoted by $\veebar$ or $\oplus$.

If so the solution would be $$\neg A \veebar \neg B$$ or without $\veebar$ you could do $$(\neg A \lor \neg B) \land \neg (\neg A \land \neg B)$$

Alternative to your solution you could do

a. Alice and Bob are not both in the room: This means when Alice is in the room then Bob isn't: $$A \to \neg B $$ This equivalent to your statement to see this just use the rule $A\to B \iff \neg A \lor B $

d. "Neither Alice nor Bob is in the room." is the negation of "Alice or Bob is in the room." So this would lead to $$\neg (A\lor B)$$ which is equivalent to the previous formula.

Nathanael Skrepek
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