Sets, Relations & Functions
Equivalence Relations on Natural Numbers
nta_pyq_2023_jan
Grade None
Question:
Let R be a relation defined on \mathbb{N} as a R b if 2a + 3b is a multiple of 5, a, b \in \mathbb{N}. Then R is
not reflexive
transitive but not symmetric
symmetric but not transitive
an equivalence relation
Step-by-Step Solution
Key Concept: Check reflexivity: 2a+3a=5a is multiple of 5 ✓. For symmetry and transitivity, use modular arithmetic.
Reflexive: 2a+3a=5a, multiple of 5. ✓ Symmetric: aRb$\Rightarrow$2a+3b=5$\alpha$, bRa$\Rightarrow$2b+3a=5(a+b−$\alpha$), multiple of 5. ✓ Transitive: aRb and bRc$\Rightarrow$2a+3b=5$\alpha$, 2b+3c=5$\beta$$\Rightarrow$2a+3c=5($\alpha$+$\beta$−b), multiple of 5. ✓ Hence R is an equivalence relation. Answer: (4)
Correct Answer: 4