_________________ _ _ (A & B) /\ (C \/ B) \/ (C & B)

To simplify the logical expression \((A \land B) \land (C \lor B) \lor (C \land B)\), we can break it down step by step using logical identities and rules. 1. **Expression to Simplify:** \[ (A \land B) \land (C \lor B) \lor (C \land B) \] 2. **Apply the Distributive Law:** Let's simplify \((A \land B) \land (C \lor B)\): \[ (A \land B \land C) \lor (A \land B \land B) = (A \land B \land C) \lor (A \land B) \] (Since \(B \land B\) is just \(B\)). 3. **Combine the Results:** Now substituting this back into the original expression: \[ (A \land B \land C) \lor (A \land B) \lor (C \land B) \] 4. **Factor Out the Common Terms:** Notice \(B\) is common in both \((A \land B)\) and \((C \land B)\): \[ B \land (A \lor C) \lor (A \land B) \] 5. **Combine Like Terms:** The expression can be simplified further: \[ (A \land B) \lor (C \land B) = B \land (A \lor C) \] Thus, the final simplified expression is: \[ B \land (A \lor C) \] In summary, the logical expression \((A \land B) \land (C \lor B) \lor (C \land B)\) simplifies to: \[ B \land (A \lor C) \]