solving iff logical statements

Question

Given that

1. Processor A reports that Processor B is not working and Processor C is working.
2. Processor B reports that Processor A is working if and only if Processor B is working.
3. Processor C reports that at least one of the other two processors is not working.

Assume that all of the status reports are true, which processor(s) is/are working?

A states that B is not working while B states A only works iff B works therefore both A and B are not working while C is the only one working.

Assuming that all of the processors are working, which status report(s) is/are false?

A is false since it implies B is not working. B is then also false since A is false. C is also false since it implies either A or B is not working

Assuming that a processor’s status report is true if and only if the processor is working, what is the status of each processor?

I am confused how to tackle this question, which processor do we begin with? how do I begin solving this?

Answer 1

For the third question, B is not working. If it did, then it would be telling the truth, hence A is working, hence A telling the truth, hence B not working, a contradiction. This shows B is not working.

Hence B not telling truth (and not working). Hence A working, hence A telling the truth, hence C working (and C telling the truth).

Answer 2

You are wrong for the second part.

Your reasoning for "B is then also false since A is false" is invalid and since $A$ and $B$ are both working, the statement by $B$ is actually true.

Answer 3

We translate each statement into the symbolic logic.

• Processor A reports $\neg B \land C$.
• Processor B reports $A\longleftrightarrow B$.
• Processor C reports $\neg A \lor \neg B$.

Here $A$ is the statement meaning "processor A is working", and similarly for the others.

The first part asks us to find which of $A$, $B$, and $C$ is true, assuming that all of the above statements are true. Your answer is correct here.

The second part asks us to find which of the above statements are false, assuming that $A$, $B$, and $C$ are true. We can just substitute $1$ (meaning truth) into the above statements and find their values:

• $\neg B \land C$ becomes $\neg 1\land 1 \equiv 0$, so processor A is not working.
• $A\longleftrightarrow B$ becomes $1\longleftrightarrow 1 \equiv 1$, so processor B is working.
• $\neg A \lor \neg B$ becomes $\neg 1 \lor \neg 1 \equiv 0$, so processor C is not working.

The third part asks us to find which of $A$, $B$, and $C$ are true, assuming the following:

• $A \longleftrightarrow (\neg B \land C)$
• $B \longleftrightarrow (A\longleftrightarrow B)$
• $C \longleftrightarrow (\neg A \lor \neg B)$

We can think of an argument using valid argument forms, but I don't see it. A brute-force approach to find the status of processor A is to compute the truth value of $$[ (A \longleftrightarrow (\neg B \land C)) \land (B \longleftrightarrow (A\longleftrightarrow B)) \land (C \longleftrightarrow (\neg A \lor \neg B))] \implies A$$ with the help of a truth-table. Since the above implication is a tautology, then our assumptions can deduce $A$ to be true. To find the status of other processors, replace the right-hand-side of the above implication by $B$ and $C$, and determine whether the implication is a tautology.