Sets, Relations & Functions
Relations — Minimum Elements for Symmetry
nta_pyq_2024_apr
Grade 11
Question:
Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3,\ldots,20\}$ be given by $R_1=\{(x,y):2x-3y=2\}$ and $R_2=\{(x,y):-5x+4y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals
Step-by-Step Solution
Key Concept: $R_1=\{(4,2),(7,4),(10,6),(13,8),(16,10),(19,12)\}$ — 6 pairs, none symmetric within the set (e.g. $(2,4)\notin R_1$). Need to add all 6 reverses $\Rightarrow M=6$. $R_2=\{(4,5),(8,10),(12,15),(16,20)\}$ — 4 pairs, reverses not in $R_2$ $\Rightarrow N=4$.
$R_1$ has 6 elements requiring 6 additions; $R_2$ has 4 elements requiring 4 additions. $M+N=10$.
Correct Answer: 4