Sets, Relations & Functions
Equivalence Relations on Power Sets
nta_pyq_2023_jan
Grade 11
Question:
Let P(S) denote the power set of S = \{1, 2, 3, \ldots, 10\}. Define the relations R_1 and R_2 on P(S) as: A R_1 B if (A \cap B^c) \cup (B \cap A^c) = \varnothing, and A R_2 B if A \cup B^c = B \cup A^c, for all A, B \in P(S). Then:
both R_1 and R_2 are equivalence relations
only R_1 is an equivalence relation
only R_2 is an equivalence relation
both R_1 and R_2 are not equivalence relations
Step-by-Step Solution
Key Concept: AR_1B ⟺ A=B (symmetric difference is empty). AR_2B ⟺ A$\cup$B^c = B$\cup$A^c ⟺ A=B. Both reduce to equality, hence both are equivalence relations.
R_1: (A$\cap$B^c)$\cup$(B$\cap$A^c)=∅ ⟺ A=B (symmetric difference = ∅). R_2: A$\cup$B^c=B$\cup$A^c ⟺ A=B. Both are equality relations on P(S), hence both are equivalence relations. Answer: (1)
Correct Answer: 1