22 votes

If $P, Q, R$ are subsets of the universal set U, then $$(P\cap Q\cap R) \cup (P^c \cap Q \cap R) \cup Q^c \cup R^c$$ is

- $Q^c \cup R^c$
- $P \cup Q^c \cup R^c$
- $P^c \cup Q^c \cup R^c$
- U

35 votes

Best answer

17 votes

Can we treat these like Boolean expression and solve?

Like PQR + P'QR + Q' + R'. and minimise this.

Is this method always correct?

@Praveen Sir?

@Arjun Sir?

0

Praveen Saini if use Aspi R Osa 's method and found P.PQ then this equivalent to PQ or we take it as P.PQ ?

0

Praveen Saini sir

https://gateoverflow.in/3562/gate2006-it-23

i above link's Ques

in I,

LHS=P+QR-PQR

RHS=(P+Q-PQ).(P+R-PR)

=P+PR-PR+PQ+QR-PQR-PQ-PQR+PQR

=P+QR-PQR

LHS=RHS

So I is true but original ans is I is false

plz verify

12 votes

3 votes

*Treating as a boolean expression like suggested in an answer here:*

*PQR + P'QR + Q' + R'*

*= (P+P') QR + Q' + R'*

*= QR + Q' + R'*

*= QR + Q'R' + Q'R + R'*

*= R(Q+Q') + R'(Q'+1)*

*= R + R'*

*= 1*

*Also R+R' means RUR' so its equal to U.*