Task on propositional logic:
"Suppose you create a truth table for A and B, both formulas in propositional calculus, and have a look at the columns below the main connectives of A and B. When do we know for sure that A ≡ B is true?"
My understanding is that this option should be correct:
"If there is a T in every row under A and a T in every row under B."
However, the answer sheet says it's not correct. Why is that?
If $A$ and $B$ have a $T$ in each row, then both of them are tautologies, and any two tautologies are logically equivalent. So, both being always True is certainly a sufficient condition for them to be equivalent. However, it is not necessary: two statements that are False in every row are also logically equivalent. And, in general, if $A$ and $B$ have the same truth-value in every row, where for some rows they may both be True, and for other rows they may both be False, then they are equivalent.
Indeed, having the same truth-value in every row is a necessary and sufficient condition: so this is exactly when they are equivalent. My guess is that this is the answer that your problem sheet was looking for. If so, I would say the question is a bit poorly phrased, since the way it is stated it does sound like your sufficient condition should work.