Question

The Boolean expression Y = \(\overline{PQ}\)R+ Q\(\bar{R}\) + \(\bar{P}\)QR + PQR simplifies to

• A. \(\bar P\)R + Q
• B. PR + \(\bar{Q}\)
• C. P+R
• D. Q+R

Answer 1

The Correct Option is D

To simplify the Boolean expression \( Y = \overline{PQ}R + Q\bar{R} + \bar{P}QR + PQR \), we will apply Boolean algebra rules. Let's break it down step by step:

1. First, we identify common terms and factorize where possible. Notice that PQR is a common sub-expression in \bar{P}QR + PQR:

\[ Y = \overline{PQ}R + Q\overline{R} + QR(\bar{P} + P) \]

2. Using the Complement Law, we know that \(\bar{P} + P = 1\). Therefore, the expression simplifies to:

\[ Y = \overline{PQ}R + Q\overline{R} + QR \]

3. Now, we analyze \(\overline{PQ}R + QR\). Notice that \(QR\) is common, so it can be factorized:

\[ Y = R(\overline{PQ} + Q) + Q\overline{R} \]

4. In the expression \(\overline{PQ} + Q\), using the Absorption Law \(A + AB = A\), we get:

\[ \overline{PQ} + Q = Q \] So the expression becomes: \[ Y = QR + Q\overline{R} \]

5. Lastly, using the Distributive Law \(A + AC = A\), combine the terms:

\[ Y = Q(R + \overline{R}) \]

6. Since \(R + \overline{R} = 1\) (Complement Law), the expression simplifies to:

\[ Y = Q \]

7. Now combine it with the remaining non-simplified segment from the beginning:

\[ Y = Q + R \]

Therefore, the simplified expression is \(Q + R\). None of the extra combinations of terms result in further simplifications or changes since all terms have been properly absorbed or simplified.

The correct answer is Q+R.