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+27 votes

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?

- (S1)-False, (S2)-True, (S3)-True, (S4)-False
- (S1)-True, (S2)-False, (S3)-False, (S4)-True
- (S1)-False, (S2)-False, (S3)-True, (S4)-True
- (S1)-True, (S2)-True, (S3)-False, (S4)-False

+30 votes

Best answer

+13 votes

$F = P{}' + QR$

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**).

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