Sets, Relations & Functions
General
Grade 11
Question:
<p>Let R1 = {(a, b) ∈N × N : |a −b| ≤13} and R2 = {(a, b) ∈N × N : |a −b| ̸= 13}. Then:</p>
Both R1 and R2 are equivalence relations
Neither R1 nor R2 is an equivalence relation
R1 is equivalence but R2 is not
R2 is equivalence but R1 is not
Step-by-Step Solution
Key Concept: Both are reflexive and symmetric (since |a -b| = |b -a|). Transitivity fails for both — use chain
counterexamples.
<p><strong>Step 1</strong>: R1: Reflexive ✓. Symmetric ✓. Transitivity fails: take a = 1, b = 14, c = 27: |1 -14| = 13 \leq13 ✓;</p><br>|14 -27| = 13 \leq13 ✓;<br>but |1 -27| = 26 > 13. ✗<p><strong>Step 2</strong>: R2: Reflexive (|a -a| = 0 ̸= 13) ✓. Symmetric ✓. Transitivity fails: take a = 1, b = 2, c = 14:</p><br>|1 -2| = 1 ̸= 13 ✓;<br>|2 -14| = 12 ̸= 13 ✓;<br>but |1 -14| = 13, so (1, 14) /\in R2. ✗
Correct Answer: 2