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Q20GATE 2003MCQ

If the function W,X,Y and Z are as follows $W=R+\overline{P}Q+\overline{R}S$ , $X=PQ\bar{R}\bar{S}+\bar{P}\bar{Q}\bar{R}\bar{S}+P\bar{Q}\bar{R}\bar{S}$ , $Y=RS+\overline{PR+P\bar{Q}+\bar{P}\bar{Q}}$ , $Z=R+S+\overline{PQ+\bar{P}\bar{Q}\bar{R}+P\bar{Q}\bar{S}}$ Then,

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📖 Explanation

$W=R+\overline{P}Q +\overline{R}S$ $x = PQ\overline{R}\;\overline{S} + \overline{P}\;\overline{Q}\;\overline{R}\;\overline{S} + P\overline{Q}\;\overline{R}\;\overline{S}$

\begin{aligned} Y &=R S+\overline{P R+P \bar{Q}+\bar{P} \bar{Q}} \\ &=R S+\overline{P R} \cdot \overline{P \bar{Q}} \overline{\bar{P} \bar{Q}} \\ &=R S+(\bar{P}+\bar{R})(\bar{P}+Q)(P+Q) \\ &=R S+(\bar{P}+\bar{P} Q+\bar{P} \bar{R}+Q \bar{R})(P+Q) \\ &=R S+\bar{P} Q+\bar{P} Q+\bar{P} Q \bar{R}+P Q \bar{R}+Q \bar{R} \\ &=R S+\bar{P} Q+Q \bar{R}(P+\bar{P})+Q \bar{R} \\ &=R S+\bar{P} Q+Q \bar{R} \end{aligned}

\begin{aligned} Z &=R+S+\overline{P Q+\bar{P} \bar{Q} \bar{R}+P \bar{Q} \bar{S}} \\ &=R+S+\overline{P Q} \cdot \overline{\bar{P} \bar{Q} \bar{R}} \cdot \overline{P \bar{Q} \bar{S}} \\ &=R+S+(\bar{P}+\bar{Q})(P+Q+R)(\bar{P}+Q+S) \\ &=R+S+(\bar{P} Q+\bar{P} R+P \bar{Q}+\bar{Q} R)(\bar{P}+Q+S) \\ &= R+S+\bar{P} Q+\bar{P} Q+\bar{P} Q S+\bar{P} R+\bar{P} Q R \\ &\qquad+\bar{P} R S+P \bar{Q}+P \bar{Q} S+\bar{P} \bar{Q} R+\bar{Q} R S \\ &= R+S+\bar{P} Q(1+S)+\bar{P} R(1+Q) \\ &\qquad+\bar{P} R S+P \bar{Q} S+\bar{P} \bar{Q} R+\bar{Q} R S \\ &= R+S+\bar{P} Q+\bar{P} R+\bar{P} R S+P \bar{Q} S+\bar{P} \bar{Q} R+\bar{Q} R S \\ &= R+S+\bar{P} Q+\bar{P} R+P \bar{Q} S+\bar{P} \bar{Q} R+\bar{Q} R S \\ &= R+S+\bar{P} Q+\bar{P} R(1+\bar{Q})+P \bar{Q} S+\bar{Q} R S \\ &=\bar{P}+\bar{S}+\bar{P} Q+\bar{P} R+P \bar{Q} S+\bar{Q} R S \end{aligned}

K $\cdot$ Map(1)= K.map (4) $\therefore W=Z$ From map (2) & (4) $X=\bar{Z}$

Explanation Figure 1

Explanation Figure 2

Explanation Figure 3