Sets, Relations & Functions
Equivalence Relation — Statements on n(R)
nta_pyq_2026_jan
Grade None
Question:
Let $A=\{0,1,2,\ldots,9\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $|x-y|$ is a multiple of 3. Given below are two statements:
**Statement I:** $n(R)=36$.
**Statement II:** $R$ is an equivalence relation.
Choose the correct answer:
Statement I is incorrect but Statement II is correct
Statement I is correct but Statement II is incorrect
Both Statement I and Statement II are incorrect
Both Statement I and Statement II are correct
Step-by-Step Solution
Key Concept: $x\equiv y\pmod3$. Equivalence classes: $\{0,3,6,9\}$ (4), $\{1,4,7\}$ (3), $\{2,5,8\}$ (3). $R$ is reflexive, symmetric, transitive $\Rightarrow$ equivalence. Statement II correct.
Statement I incorrect, Statement II correct.
Correct Answer: 1