If A → ¬B and B → ¬A, what can be concluded about A and B?

• A. They must both be true
• B. They must both be false
• C. A and B are equivalent
• D. They cannot both be true; at most one is true

Answer: (D)

If A is true, then B must be false, and if B is true, then A must be false. That means A and B cannot both be true at the same time. So at most one of them can be true.

All four truth-value combinations are allowed except the one where both are true. For example, both false works, or exactly one of them true works, but having both true leads to a contradiction with the implications. This is why the conclusion is that they cannot both be true; at most one is true.

They are not equivalent, because there are scenarios where A is true and B is false or B is true and A is false. Equivalence would require them to share the same truth value in all cases, which isn’t the case here.