Let P, Q, and R be sets. Let delta denote the symmetric difference operator defined as P delta Q = (P ∪ Q) - (P ∩ Q). Using Venn diagrams, determine which of the following is/are TRUE?
I. P delta (Q ∩ R) = (P delta Q) ∩ (P delta R)
II. P delta (Q ∪ R) = (P delta Q) ∪ (P delta R)
A. I only
B. II only
C. Neither I nor II
D. Both I and II
GATE 2006 · Discrete Mathematics · Set Theory · medium
Answer: Neither Statement I nor Statement II is true in general. Answer: C.
Test Statement I with a counterexample: Let P = {1,2}, Q = {1,3}, R = {1,4}. Then Q∩R = {1}. LHS = P△{1} = {2} (since {1} is in P but not {2}, and {1} cancels). (P△Q) = {2,3}, (P△R) = {2,4}. RHS = {2,3}∩{2,4} = {2}. This case agrees, so try P = {1,2}, Q = {2,3}, R = {3,4}. Q∩R = {3}. LHS = {1,2}△{3} = {1,2,3}. P△Q = {1,3}, P△R = {1,2,3,4}. RHS = {1,3}∩{1,2,3,4} = {1,3}. LHS = {1,2,3} ≠ RHS = {1,3}. Statement I is FALSE.
Test Statement II with a counterexample: Let P = {1,2}, Q = {2,3}, R = {3,4}. Q∪R = {2,3,4}. LHS = {1,2}△{2,3,4} = {1,3,4}. P△Q = {1,3}, P△R = {1,2,3,4}. RHS = {1,3}∪{1,2,3,4} = {1,2,3,4}. LHS = {1,3,4} ≠ RHS = {1,2,3,4}. Statement II is FALSE.