GATE CSE 2008 | Question: 2

Question

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

• A. $Q^c \cup R^c$
• B. $P \cup Q^c \cup R^c$
• C. $P^c \cup Q^c \cup R^c$
• D. $\cup$

Comments

convert to Boolean expr then do

Answer 1

Answer D

$\quad(P\cap Q\cap R)\cup (P^{c}\cap Q\cap R)\cup Q^{c}\cup R^{c}$

$=(P\cup P^{c})\cap (Q\cap R)\cup Q^{c}\cup R^{c}$

$=(Q\cap R)\cup Q^{c}\cup R^{c}$

$=(Q\cap R)\cup (Q\cap R)^{C}$

$= U.$

Answer 2

so option d 

Comments

this explanation made it so easy. thanks.....

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$\cap\equiv $AND  gate

$\cup\equiv $ OR  gate 

 

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best method!!

Answer 3

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?

Comments

$A-B = A \cap B'$
$P\Delta (Q\cap R)$= P-(Q.R) = P.(QR)' = PQ'+PR' that is $(P\Delta Q) \cup (P\Delta R)$

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 Praveen Saini  sir 

whats wrong in my explanation 

plz verify 

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@Praveen Saini sir, i think the $\Delta$ notation represents symmetric difference. but, you have used it here in $P \Delta (Q \cap R)$ as set difference. it should be, i think $P - (Q \cap R)$.

Answer 4

hope it might help....

Comments

But the problem with this solution is " the diagram"! How did u come to the conclusion that the diagram looks like the one you have drawn ? They haven't said anything Abt their intersection right?  All three can be independent sets and still be a subset of U!

Do correct me if wrong:)

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IF THEIR IS NO INTERSECTION THE VALUE IN THE INTERSECTION BOX OF THE DIAGRAM WILL COME OUT TO BE ZERO AUTOMATICALLY.

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@Doodle) It's because it cover's all the possibilities that the 3 sets can cover.

Answer 5

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.

Answer 6

If someone is good in digital Logic part or in vein Diagram part, than this question is easy for them :)

Just convert Union into + and intersection in . and try to solve it.

else Vein diagram becomes very easy for understanding.

If Understood UpVoted :)