Sets, Relations & Functions
General
Grade 11
Question:
<p>Let A = A1 ∪A2 ∪· · · ∪Ak where Ai ∩Aj = ∅for i ̸= j. Define R = {(x, y) : y ∈Ai ⇔
x ∈Ai}. Then R is:</p>
Reflexive and symmetric but not transitive
Reflexive and transitive but not symmetric
Reflexive but not symmetric and transitive
An equivalence relation
Step-by-Step Solution
Key Concept: x and y are related iff they belong to the same block Ai. Every partition ↔equivalence relation —
one of the most fundamental theorems in set theory.
<p><strong>Step 1</strong>: Reflexive: Every x belongs to some Ai; x \in Ai \Leftrightarrow x \in Ai trivially. ✓</p><p><strong>Step 2</strong>: Symmetric: (x, y) \in R means x, y \in Ai for the same i — this condition is symmetric in x and y. ✓</p><p><strong>Step 3</strong>: Transitive: (x, y) \in R: x, y \in Ai. (y, z) \in R: y, z \in Aj. Since Ai \capAj = \emptysetfor i ̸= j and y \in Ai \capAj, we</p><br>must have i = j. So x, z \in Ai, giving (x, z) \in R. ✓
Correct Answer: 4