Multiple choice

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

  1. $W = Z, X = \bar Z$
  2. $W = Z, X = Y$
  3. $W = Y$
  4. $W = Y = \bar Z$
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A Correct answer
Explanation

$W = R + \bar PQ + \bar RS$ $X = PQ\bar R\bar S+\bar P\bar Q\bar R\bar S+ \bar P\bar Q\bar R\bar S$ $Y = RS + \overline{PR + P\bar Q + \bar P \bar Q} = RS + \bar PR.\overline{PR}.\overline{P\bar Q}\overline{\bar P\bar Q}=RS + (\bar P + bar R)(\bar P + Q)(P + Q) = RS + (\bar P + \bar RQ)(P +Q) = RS + \bar pQ + \bar RPQ + Q\bar R$

$Z=R+S+\overline{PQ+\bar P.\bar Q.\bar R + P\bar Q.\bar S}=R+S+\overline{PQ}.\overline{\bar p\bar Q\bar S} = R + S + (\bar P + Q)(P+Q+R).(\bar P + Q + S)= R + S+ (\bar P + \bar Q(Q +S)).(P+ Q+ R)= R +S +\bar P Q + \bar P R+\bar QPS + \bar QRS(P + Q + R) . (\bar P + Q + S)= R + S + (\bar P + \bar Q (Q + S)) . (P + Q + R)= R + S + \bar P Q + \bar PR + \bar PPS + \bar PRS $

From above, we can conclude that $W = Z, X =\bar Z$