search
Log In
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
22 votes
3.9k views

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
in Set Theory & Algebra
edited by
3.9k views

7 Answers

35 votes
 
Best answer

Answer D

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

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

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

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

$= U.$


edited by
23 votes

so option d 

0
this explanation made it so easy. thanks.....
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?

15
Yes absolutely correct , will get 1 , that is U
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
Yes it will be 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

2
$A-B = A \cap B'$
$P\Delta (Q\cap R)$= P-(Q.R) = P.(QR)' = PQ'+PR' that is $(P\Delta Q) \cup (P\Delta R)$
0

 Praveen Saini  sir 

whats wrong in my explanation 

plz verify 

12 votes

hope it might help....

0
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:)
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.

0 votes

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 :)

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

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

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

$\bigcup$                                                                                            {$A \cup A’$ = $U$}
ago
Answer:

Related questions

30 votes
3 answers
1
4.7k views
Which one of the following is false? The set of all bijective functions on a finite set forms a group under function composition The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod $p$, where $p$ is a prime number The set of all strings over a finite alphabet forms a group ... the group $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$
asked Oct 9, 2014 in Set Theory & Algebra Kathleen 4.7k views
56 votes
5 answers
2
8k views
Consider the following relation on subsets of the set $S$ of integers between $1$ and $2014$. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum element in the symmetric difference of the two sets is in $U$. Consider the following two statements: $S1$: There ... $S2$ are true $S1$ is true and $S2$ is false $S2$ is true and $S1$ is false Neither $S1$ nor $S2$ is true
asked Sep 28, 2014 in Set Theory & Algebra jothee 8k views
37 votes
8 answers
3
4.4k views
Consider the following statements: $S_1:$ There exists infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite. $S_2:$ There exists two irrational numbers $x$ and y such that $(x+y)$ is rational. Which of the following is true about $S_1$ and $S_2$? Only $S_1$ is correct Only $S_2$ is correct Both $S_1$ and $S_2$ are correct None of $S_1$ and $S_2$ is correct
asked Sep 14, 2014 in Set Theory & Algebra Kathleen 4.4k views
19 votes
8 answers
4
6.6k views
The number of integers between $1$ and $500$ (both inclusive) that are divisible by $3$ or $5$ or $7$ is ____________ .
asked Feb 14, 2017 in Set Theory & Algebra Arjun 6.6k views
...