Given the function $F = P' +QR$, where $F$ is a function in three Boolean variables $P, Q$ and $R$ and $P'=!P$, consider the following statements.
$(S1) F = \sum(4, 5, 6)$
$(S2) F = \sum(0, 1, 2, 3, 7)$
$(S3) F = \Pi (4, 5, 6)$
$(S4) F = \Pi (0, 1, 2, 3, 7)$
Which of the following is true?
P ′ = ! P means ??
$F = P{}' + QR$
for SOP we have :
$F = P{}.1.1 + 1.QR'=P{}'(Q+Q{}')(R+R{}') + (P+P{}')QR$
$P{}'QR + P{}'QR{}' +P{}'Q{}'R+P{}'Q{}'R{}'+PQR+P{}'QR$
$P{}'QR + P{}'QR{}' +P{}'Q{}'R+P{}'Q{}'R{}'+PQR$
$F=\sum(0, 1, 2, 3, 7)$ (considering barred terms as 0 and unbarred as 1 and converting them to binary and then to decimal).
now for POS we have :
$F = P{}' + QR = (P{}'+Q)(P{}'+R) = (P{}'+Q+0)(P{}'+R+0)$
$(P{}'+Q+R.R{}')(P{}'+R+Q.Q{}') $
$(P{}'+Q+R)(P{}'+Q+R)(P{}'+Q+R{}')(P{}'+Q{}'+R)$
$(P{}'+Q+R)(P{}'+Q+R{}')(P{}'+Q{}'+R)$
$F=\prod (4, 5, 6)$(considering barred terms as 1 and unbarred as 0 and converting them to binary and then to decimal).
http://mcs.uwsuper.edu/sb/461/PDF/sop.html
Answer = A
S2 and S3 are true
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