Is The converse of a biconditional statement is always true?

Answer 1

No, not always. It depends on if the original biconditional statement is true. For example take the following biconditional statement:

x = 3 if and only if x2 = 9.

From this biconditional statement we can extract two conditional statements (hence why it is called a bicondional statement):

The Conditional Statement: If x = 3 then x2 = 9.

This statement is true. However, the second statement we can extract is called the converse.

The Converse: If x2=9 then x = 3.

This statement is false, because x could also equal -3. Since this is false, it makes the entire original biconditional statement false.

All it takes to prove that a statement is false is one counterexample.