Within Intermediate Logic Lesson 11, what would keep us from setting up the propositions both being true at the same time, and if there were a contradiction they would not be equivalent? Instead of setting them up one true and one false and if there’s a contradiction then they are equivalent?
That would be checking for consistency, not equivalence. If you set them both as true, and get a contradiction, then they are not consistent (which of course also means they are not equivalent, nor related by implication, per the chart in Introductory Logic, p. 71). But if you get no contradiction, all you have shown is that they can both be true, which is the meaning of consistency. To show equivalence, you have to show that they cannot have opposite truth values: the first cannot be true while the second is false, and vice versa.