Sets, Relations & Functions
Relations — Size and Symmetrising
nta_pyq_2024_apr
Grade 11
Let $A=\{1,2,3,4,5\}$. Let $R$ be a relation on $A$ defined by $x\,R\,y$ if and only if $4x\leq5y$. Let $\mathrm{m}$ be the number of elements in $R$ and $\mathrm{n}$ be the minimum number of elements from $A\times A$ that are required to be added to $R$ to make it a symmetric relation. Then $\mathrm{m}+\mathrm{n}$ is equal to:
Step-by-Step Solution
Key Concept: List all $(x,y)\in A\times A$ with $4x\leq5y$: includes $(1,1),(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5),(4,4),(4,5),(5,4),(5,5)$ — total 16. So $\mathrm{m}=16$.
$\mathrm{m}=16$ (elements in $R$), $\mathrm{n}=9$ (additions needed). $\mathrm{m}+\mathrm{n}=25$.
Correct Answer: 1