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

  1. $PΔ (Q ∩ R) = (P Δ Q) ∩ (P Δ R)$
  2. $P ∩ (Q ∩ R) =  (P ∩ Q) Δ (P Δ R)$
  1. I only
  2. II only
  3. Neither I nor II
  4. Both I and II
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4 Answers

Best answer
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36 votes

Answer C.

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………………......………………………………………….

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Best Method

 

The symmetric difference using Venn diagram of two subsets A and B is a subset of U, denoted by A △ B and is defined by

A △ B = (A – B) ∪ (B – A) = AB’ + BA’

Let A and B are two sets. The symmetric difference of two sets A and B is the set (A – B) ∪ (B – A) and is denoted by A △ B.

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The correct option is B 

PΔQ=PQ′+P′Q
where Δ is symmetric difference between P and Q
I. LHS =PΔQ∩R=PΔ(QR)=P(QR)′+P′(QR)
=P(Q′+R′)+P′QR
RHS =(PΔQ)∩(PΔR)=(PQ′+P′Q)(PR′+P′R)=PQ′R′+P′QR
LHS ≠ RHS
So statement I is false.
II. LHS =P∩QΔR=P(QR′+Q′R)
=PQR′+PQ′R
RHS =(P∩Q)Δ(P∩R)=PQΔPR=PQ(PR)′+(PQ)′PR=PQ(P′+R′)+(P′+Q′)PR
=PQR′+PQ′R
LHS = RHS
So statement II is true.

 

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