Let P, Q and R be sets let Δ denote the symmetric difference operator defined…
2006
Let P, Q and R be sets let Δ denote the symmetric difference operator defined as PΔQ = (P U Q) - (P ∩ Q). Using Venn diagrams, determine which of the following is/are TRUE?
PΔ (Q ∩ R) = (P Δ Q) ∩ (P Δ R)
P ∩ (Q ∩ R) = (P ∩ Q) Δ (P Δ R)
- A.
1 only
- B.
2 only
- C.
Neither 1 nor 2
- D.
Both 1 and 2
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Correct answer: C
Final answer: Neither 1 nor 2.
Both given equalities are false. Below are concise counterexamples demonstrating each failure.
Statement 1: P Δ (Q ∩ R) = (P Δ Q) ∩ (P Δ R) is false. Counterexample: take P = {1}, Q = {1}, R = ∅. Then Q ∩ R = ∅, so P Δ (Q ∩ R) = P Δ ∅ = P = {1}. Meanwhile, P Δ Q = P Δ P = ∅ and P Δ R = P Δ ∅ = P, so (P Δ Q) ∩ (P Δ R) = ∅ ∩ P = ∅. Since {1} ≠ ∅, the equality fails.
Statement 2: P ∩ (Q ∩ R) = (P ∩ Q) Δ (P Δ R) is false. Counterexample: take P = {1}, Q = ∅, R = ∅. Then P ∩ (Q ∩ R) = {1} ∩ (∅ ∩ ∅) = ∅. But P ∩ Q = ∅ and P Δ R = {1} Δ ∅ = {1}, so (P ∩ Q) Δ (P Δ R) = ∅ Δ {1} = {1}. Since ∅ ≠ {1}, the equality fails.
Therefore neither equality holds in general, so the correct choice is the option that states neither 1 nor 2 is true.
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