Let A and B be two propositions. "A if and only if B" means that A is true if B is true and that A is true only if B is true. Can you please intuitively explain why "A if and only if B" implies that A and B have the same truth value? I say "intuitively" because one can prove this formally by a truth table. But I want to really understand it. For example, can you give enlightining examples?
To say that A if and only if B is to say that A and B are logically equivalent, i.e. they always have the same truth value. That's because whenever A is true, B is also true (A only if B), and whenever B is true, A is also true (A if B). So A and B are true in exactly the same cases, and false in exactly the same cases.
Here's a simple example. Say the following is true:
It's snowing if and only if it's winter.
It follows that the days when it's snowing and the days when it's winter are exactly the same days. Because for each day, if it's snowing then it also has to be winter (it's snowing only if it's winter), and if it's winter then it's also snowing (it's snowing if it's winter).