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

To simplify the expression \((A \land B) \lor (C \lor B) \lor (C \land B)\), we can apply logical identities. 1. Recall that: - \(\land\) represents logical AND - \(\lor\) represents logical OR Let's break it down step by step: 1. **Expand the expression:** \((A \land B) \lor (C \lor B) \lor (C \land B)\) 2. Using the distributive property, we recognize that \((C \lor B)\) can be simplified as follows: - \(C \lor B\) is already in its simplest form, so we can just include both terms. 3. Rearranging the expression, we distribute \(B\) across \((C \land B)\): \((A \land B) \lor (C \lor B) \lor (C \land B)\) can be rearranged to: \[ = (A \land B) \lor ((C \land B) \lor C) \lor B \] 4. Now using the Absorption Law, we can simplify it: - \(C \lor (C \land B) = C\) since \(C\) covers both cases. This leads us to: \[ = (A \land B) \lor C \lor B \] 5. Finally, we can see if \(B\) can be absorbed: - The expression becomes \((A \land B) \lor C \lor B = (C \lor B)\) since \(B\) is common. Thus, the entire expression can be simplified to: \[ C \lor B \] This means that the original expression simplifies down to: \[ B \lor C \] In summary, \[ (A \land B) \lor (C \lor B) \lor (C \land B) \equiv B \lor C \]