Sets, Relations & Functions
General
Grade 11
Question:
<p>Let P(S) be the power set of S = {1, 2, . . . , 10}. Define A R1 B iff (A ∩Bc) ∪(B ∩Ac) = ∅,
and A R2 B iff A ∪Bc = B ∪Ac. Then:</p>
Both R1 and R2 are equivalence relations
Only R1 is an equivalence relation
Only R2 is an equivalence relation
Neither R1 nor R2 is an equivalence relation
Step-by-Step Solution
Key Concept: (A \capBc) \cup(B \capAc) = A\triangleB (symmetric difference). A\triangleB = \emptyset\Leftrightarrow A = B. Similarly A \cupBc =
B \cupAc \Leftrightarrow A = B. Both reduce to set equality.
<p><strong>Step 1</strong>: R1: (A \capBc) \cup(B \capAc) = A\triangleB. Since A\triangleB = \emptyset\Leftrightarrow A = B, R1 is the equality relation — trivially an</p><br>equivalence. ✓<p><strong>Step 2</strong>: R2: A \cupBc = B \cupAc. Complementing both sides gives Ac \capB = Bc \capA, i.e. B \ A = A \ B. This forces</p><br>A \ B = B \ A = \emptyset, i.e. A = B. R2 is also the equality relation. ✓
Correct Answer: 1