A <->B = (A v B) & (B v A)

The expression you provided is a logical equivalence in propositional logic. Let's break it down step by step.

1. Understanding the Symbols:

   • \( A \) and \( B \) are propositional variables.
   • \( \leftrightarrow \) (or <->) denotes bi-conditional, meaning "A if and only if B".
   • \( v \) represents the logical "or" (disjunction).
   • \( & \) represents the logical "and" (conjunction).

2. Interpretation of the Bi-conditional:

   • The statement \( A \leftrightarrow B \) can be understood as:
     • If \( A \) is true, then \( B \) is true.
     • And if \( B \) is true, then \( A \) is true.
   • This is equivalent to saying that both \( A \) and \( B \) are either both true or both false.

3. Breaking Down the Expression:

   • \( (A \lor B) \) means "A or B is true".
   • \( (B \lor A) \) is logically the same as \( (A \lor B) \).
   • The expression \( (A \lor B) \land (B \lor A) \) simplifies to just \( (A \lor B) \) because both parts are the same.

4. Examining the Logic:

   • For \( A \leftrightarrow B \) to hold true, either:
     • Both \( A \) and \( B \) are true (\( A = true, B = true \)) or
     • Both \( A \) and \( B \) are false (\( A = false, B = false \)).
   • Now, let's see when \( (A \lor B) \) is true:
     • It is true if at least one of \( A \) or \( B \) is true.
   • However, \( (A \lor B) \) will not be true when both \( A \) and \( B \) are false.

5. Conclusion:

   • The expression \( (A \lor B) \land (B \lor A) \) does not capture the necessary relationship of the bi-conditional.
   • The proper expression for \( A \leftrightarrow B \) should instead be \( (A \land B) \lor (\neg A \land \neg B) \) which means both are true or both are false.

So to clarify, the statement \( A \leftrightarrow B = (A \lor B) \land (B \lor A) \) is incorrect as it does not accurately represent the bi-conditional relationship between \( A \) and \( B \). The correct formulation for bi-conditional is \( (A \land B) \lor (\neg A \land \neg B) \).