Sets, Relations & Functions
Properties of Relations on R
nta_pyq_2023_jan
Grade 11
Question:
Let R be a relation on \mathbb{R}, given by R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number}\}. Then R is
Reflexive but neither symmetric nor transitive
Reflexive and transitive but not symmetric
Reflexive and symmetric but not transitive
An equivalence relation
Step-by-Step Solution
Key Concept: Check reflexivity: 3a−3a+$\sqrt$7=$\sqrt$7 (irrational) ✓. For symmetry, find a counterexample; for transitivity, find a counterexample.
Reflexive: 3(a−a)+$\sqrt$7=$\sqrt$7 is irrational ✓. Symmetric: a=$\sqrt$7/3, b=0: (a,b)$\in$R but 3(0)−3($\sqrt$7/3)+$\sqrt$7=0, rational, so (b,a)$\notin$R ✗. Transitive: (a,b)=( $\sqrt$7/3,1) and (b,c)=(1,2$\sqrt$7/3): both in R but (a,c)$\notin$R ✗. Answer: (1)
Correct Answer: 1