GATE CS 2008 Q26 — Boolean Algebra

We can simplify the given Boolean expression step-by-step:

Let the expression be $E = (P + \bar{Q}) (P\bar{Q} + PR) (\bar{P}\bar{R} + \bar{Q})$.

Step 1: Simplify the product of the first two terms
Using the distributive law, we can factor out $P$ from the second term: $$(P + \bar{Q}) (P\bar{Q} + PR) = (P + \bar{Q}) \cdot P(\bar{Q} + R)$$ By the absorption law, $P(P + \bar{Q}) = P$. Thus: $$P(P + \bar{Q})(\bar{Q} + R) = P(\bar{Q} + R) = P\bar{Q} + PR$$

Step 2: Multiply with the third term
Now, multiply this result with the third term $(\bar{P}\bar{R} + \bar{Q})$: $$E = (P\bar{Q} + PR)(\bar{P}\bar{R} + \bar{Q})$$ Expanding the terms: $$E = P\bar{Q}\bar{P}\bar{R} + P\bar{Q}\bar{Q} + PR\bar{P}\bar{R} + PR\bar{Q}$$ Since $P \cdot \bar{P} = 0$ and $R \cdot \bar{R} = 0$, the terms $P\bar{Q}\bar{P}\bar{R}$ and $PR\bar{P}\bar{R}$ simplify to $0$: $$E = 0 + P\bar{Q} + 0 + P\bar{Q}R$$ Applying the absorption law again: $$E = P\bar{Q}(1 + R) = P\bar{Q}$$

Therefore, the expression simplifies to $P.\bar{Q}$.