Question

Let $P,Q$ and $R$ be sets let $\Delta$ denote the symmetric difference operator defined as $P\Delta Q=(P\cup Q)-(P\cap Q)$. Using Venn diagrams, determine which of the following is/are TRUE?
I. $P\Delta (Q\cap R)=\left(P\Delta Q\right)\cap \left(P\Delta R\right)$
II. $P\cap \left(Q\Delta R\right)=(P\cap Q)\Delta (P\cap R)$

• A. I only
• B. II only
• C. Neither I nor II
• D. Both I and II
  

Answer 1

The correct option is B II only

$P\Delta Q=P{Q}^{{}^{\prime }}+P^{{}^{\prime }}Q$
where $\Delta$ is symmetric difference between $P$ and $Q$
I. LHS $=P\Delta Q\cap R=P\Delta \left(QR\right)=P(QR{)}^{{}^{\prime }}+P^{{}^{\prime }}(QR)$
$=P(Q^{{}^{\prime }}+R^{{}^{\prime }})+P^{{}^{\prime }}QR$
RHS $=\left(P\Delta Q\right)\cap \left(P\Delta R\right)=(P{Q}^{{}^{\prime }}+P^{{}^{\prime }}Q)(P{R}^{{}^{\prime }}+P^{{}^{\prime }}R)=P{Q}^{{}^{\prime }}{R}^{{}^{\prime }}+P^{{}^{\prime }}QR$
LHS $\ne$ RHS
So statement I is false.
II. LHS $=P\cap Q\Delta R=P(Q{R}^{{}^{\prime }}+Q^{{}^{\prime }}R)$
$=PQ{R}^{{}^{\prime }}+P{Q}^{{}^{\prime }}R$
RHS $=(P\cap Q)\Delta (P\cap R)=PQ\Delta PR=PQ(PR{)}^{{}^{\prime }}+(PQ{)}^{{}^{\prime }}PR=PQ(P^{{}^{\prime }}+R^{{}^{\prime }})+(P^{{}^{\prime }}+Q^{{}^{\prime }})PR$
$=PQ{R}^{{}^{\prime }}+P{Q}^{{}^{\prime }}R$
LHS = RHS
So statement II is true.