I am struggling with this question from my discrete mathematics class. I am just not sure how approach this problem. You're only supposed to use truth tables to solve it.
Here is the problem: "In an islander problem, we are given a set of statements that are about one another. Each of a set of islanders makes a statement about the truth or falsity of their own or other islanders’ statements. Each statement is either true or false, so we have a set of compound propositions with an atomic variable for each. Normally the problem is set up so that exactly one setting of the variables matches the variables. That is, each statement whose variable is set to true becomes true, and each one set to false becomes false. Given the following situations, which islanders, if any, are telling the truth?"
a) "There are two islanders. A says “Exactly one of us is lying”. B says “If A is telling the truth, then so am I."
Here is what I've tried:
I first wrote the two compound propositions given:
A ↔ (¬A ∨ ¬B) B ↔ (A → B)
Next, I created a truth table for both propositions under the assumption that if A or B is telling the truth then when either of them are true, their statements should also be true, and when either of them are false their statements should also be false.
However, after doing this I realize that this is probably not the correct way. I feel like I'm misunderstanding something. If someone could please help to lead me in the correct direction that would be really great, thanks.