This was my first time hearing about Truth Tables. Thanks to you, I’ve become quite the expert on them (-quite) (see what I did there), but I have to say, it was a fun little rabbit hole.

Anyways, as for the question itself:

p (the proposition): (Av(-B)) ↔ (B)

As you know, (-B) means (not B) and it’s basically the opposite values of (B), so we’ll just have to turn every T into F and vice versa.

So, (-B) becomes:

F

T

F

T

Now, the first part of the proposition is: A v (-B)

v means or, which means that, *AT LEAST* one of them has to be true for the overall value to be true.

(A) (-B) (Av(-B))

T F T

T T T

F F F

F T T

Lastly, we have one last operation which is: ↔

this means: If and only if

so basically, (Av(-B)) if and only if (B)

to solve this we just need to look for two similar values for the overall value to be true, if they’re not, then false it is.

We begin by bringing over the coloumn we solved earlier and compare it to (B). The answer to this is going to be P.

(Av(-B)) (B) (P which is (Av(-B)) ↔ (B))

T T T

T F F

F T F

T F F

And now you have all the pieces to the puzzle. The choices include A, B, and P so we’ll just go ahead and write those down:

A B P

T T T

T F F

F T F

F F F

Therefore, the correct choice is indeed A.

EDIT: things got a little crammed up in there, so I tried to fix it.

EDIT2: OH NO