Sets, Relations & Functions
Properties of Relations on N×N
nta_pyq_2023_jan
Grade 11
Question:
Let R be a relation on \mathbb{N} \times \mathbb{N} defined by (a, b) R (c, d) if and only if ad(b - c) = bc(a - d). Then R is
symmetric but neither reflexive nor transitive
transitive but neither reflexive nor symmetric
reflexive and symmetric but not transitive
symmetric and transitive but not reflexive
Step-by-Step Solution
Key Concept: Check each property: for reflexive, (a,b)R(a,b)$\Rightarrow$ab(b-a)=ba(a-b); for symmetric check if (a,b)R(c,d)$\Rightarrow$(c,d)R(a,b); for transitive check with counterexample.
Symmetric: (a,b)R(c,d)$\Rightarrow$ad(b−c)=bc(a−d); (c,d)R(a,b)$\Rightarrow$cb(d−a)=da(c−b)=−ad(b−c)=−bc(a−d)=cb(d−a) ✓. Reflexive: ab(b−a)$\neq$ba(a−b) in general ✗. Transitive: counterexample shows not transitive. Answer: (1)
Correct Answer: 1