translating logical quantifiers

Question

Let a = “A is working,” b = “B is working,” and c = “C is working.” Write the three status reports in terms of a, b, and c, using the symbols of formal logic.

Processor A reports that Processor B is not working and Processor C is working.

a = ~B ^ C

Processor B reports that Processor A is working if and only if Processor B is working.

b = A iff B

Processor C reports that at least one of the other two processors is not working

c = (~∀x)Wx

I need to express a,b,c using a truth table but I am confused as how to express c as it does not include variables a nor b. Is there another way to write c? or have I made a mistake?

Answer 1

You already know that

1. "B is not working" translates to $\neg B$
2. "at least one" translates to "A or B"

Just piece it together. There's no need for quantifiers.

Answer 2

Processor C reports that at least one of the other two processors is not working

I am confused as how to express c as it does not include variables a nor b. Is there another way to write c? or have I made a mistake?

To restate Processor C's report: "either : A is not working, or B is not working". Can you write that?

Answer 3

You don't need quantifiers at all, you just need to write what the status reports are. They are:

1. $\neg b \land c$
2. $a\leftrightarrow b$
3. $\neg a\lor \neg b$.