Sets, Relations & Functions
Relations — Equivalence / Properties on Product Set
nta_pyq_2024_apr
Grade 11
Question:
Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A\times B$ defined by $(a,b)\,R\,(c,d)$ if and only if $3ad-7bc$ is an even integer. Then the relation $R$ is
an equivalence relation.
reflexive and symmetric but not transitive.
transitive but not symmetric.
reflexive but not symmetric.
Step-by-Step Solution
Key Concept: Reflexive: $(a,b)R(a,b)\Rightarrow3ab-7ba=-4ab$, always even ✓. Symmetric: if $3ad-7bc$ is even, then $3cb-7da=-(3ad-7bc)+(3cb+3ad-7bc-7da)$... check parity cases: (odd,odd) or (even,even) pairs give even in both directions ✓.
$R$ is reflexive and symmetric but not transitive.
Correct Answer: 2