Doubt in logical equivalence

Question

I am beginner in mathematical logic. There is a statement in textbook(Discrete maths by Rosen) that for logical equivalence of 2 predicates, we need to show that both have same truth values(same true and false values right?).But in example problems , (say$ A \equiv B$ ), A and B are logically equivalent when

True:

1) If A is true then B is true

2)If B is true then A is true

False:(in example problems he is not considering this. why?)

3) If A is false then B is false

4)If B is false then A is false

Can you pls clarify above doubt? Also pls suggest any specific textbook on mathematical logic(simple and easy to understand textbook will be of great help).

Answer 1

Asserting “If A is true then B is true” is equivalent to asserting that “If B is false then A is false” and asserting “If B is true then A is true” is equivalent to asserting that “If A is false then B is false”.

Answer 2

Suppose B is false. Statement $1)$ implies that A cannot be true so we can deduce statement $4)$ from statement $1)$. You can do statement $3)$.