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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

1. $Q^c \cup R^c$
2. $P \cup Q^c \cup R^c$
3. $P^c \cup Q^c \cup R^c$
4. U

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

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

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

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

$= U.$

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so option d

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### 1 comment

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

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?

Yes absolutely correct , will get 1 , that is U

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 ?

Yes it will be PQ

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

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

whats wrong in my explanation

plz verify

hope it might help....

### 1 comment

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:)

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.

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 :)

$(p \cap q \cap r) \cup (p' \cap q \cap r) \cup q' \cup r’$

$(p \cap q \cap r) \cup ((p' \cap q \cap r)' \cap q \cap r)'$                          {demorgan’s law}

$(p \cap q \cap r) \cup (p \cap q \cap r)'$                                  {after solving the second part}

$\bigcup$                                                                                            {$A \cup A’$ = $U$}
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