_________________ _ _ (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.
- Recall that:
- \(\land\) represents logical AND
- \(\lor\) represents logical OR
Let's break it down step by step:
Expand the expression:
\((A \land B) \lor (C \lor B) \lor (C \land B)\)
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.
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:
\[ = (A \land B) \lor ((C \land B) \lor C) \lor B \]
Now using the Absorption Law, we can simplify it:
- \(C \lor (C \land B) = C\) since \(C\) covers both cases.
This leads us to:
\[ = (A \land B) \lor C \lor B \]
- 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, \[ (A \land B) \lor (C \lor B) \lor (C \land B) \equiv B \lor C \]
