[Math] Evaluating Logical Statements

logicproof-verificationpropositional-calculus

The problem I am working on is, "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.

(c) Either Alice or Bob is not in the room.

(d) Neither Alice nor Bob is in the room."

Let $A=$ "Alice is in the room", and let $B=$ "Bob is in the room."

(a): I know it is saying that only person is in the room; that two can't occupy it as once. I tried to represent the statement with propositional variables and logic: $(A \vee B)\wedge \neg(A\wedge B)$, this doesn't appear correct, though.

(b): Would the answer be $\neg(A\wedge B)$?

(c)(d) Would the same answer apply to both (c) and (d), namely, $\neg(A\vee B)$?

Best Answer

For (a)

(a) Alice and Bob are not both in the room.

You have: $(A \vee B)\wedge \neg(A\wedge B)$

This asserts that Alice is in the room, or Bob is in the room, but not both are in the room. Or as you put it: "I know...that two can't occupy it as once."

What you don't know is that either Alice or Bob is actually in the room, only that they are not both in the room. Put differently, you know that at most one of Alice, Bob is in the room, or that at least one is not in the room. "A is not in the room, or B is not in the room". So we only need the right conjunct of your answer:

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

(b) Alice and Bob are both not in the room $\iff$ Neither Alice nor Bob is in the room $\iff$ Alice is not in the room, and Bob is not in the room.

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

  • Your answer for (b): $\lnot(A \land B)$ is equivalent to $\lnot A \lor \lnot B$ by DeMorgan's. So what your answer asserts "Alice is not in the room, or Bob is not in the room." It's a translation of (a), but not (b).

(c) is equivalent to (a): "either not A, or not B" or "not (both A and B)":

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

  • Understandably, since natural language lends itself to ambiguity, it may seem that (c) is asserting that one of the two is in the room, but not both. That's often what people mean when they use "either....or (else)...". But in this interpretation, to avoid ambiguity, (c) would then read "Either Alice is in the room and Bob is not, or else Bob is in the room, and Alice is not," in which case you'd have $(A \lor B) \land \lnot(A \land B).$

(d) is equivalent to (b): $$\lnot A \land \lnot B \iff \lnot(A \lor B)$$

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