Let $A=\{0,1,2,3,4,5\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if max $\{x, y\}…

Let $A=\{0,1,2,3,4,5\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if max $\{x, y\} \in\{3,4\}$. Then among the statements $\left(\mathrm{S}_1\right)$ : The number of elements in R is 18 , and $\left(\mathrm{S}_2\right)$ : The relation R is symmetric but neither reflexive nor transitive
  1. both are true
  2. both are false
  3. only $\left(\mathrm{S}_2\right)$ is true
  4. only $\left(\mathrm{S}_1\right)$ is true

Solution

$A=\{0,1,2,3,4,5\}$
$\mathrm{R} \equiv\{(0,3),(3,0),(0,4),(4,0),(1,3),(3,1),(1,4)$,
$(4,1),(2,3),(3,2),(2,4),(4,2),(3,3),(3,4),(4,3)$,
$(4,4)\}$
Total 16 elements
Not reflexive as $(0,0), \ldots \ldots,(2,2) \notin \mathrm{R}$
Symmetric $\because \forall$ all a,b
$(a, b) \&(b, a) \in R$
Not transitive $\because(0,3),(3,1) \in R$
but $(0,1) \notin \mathrm{R}$
$\Rightarrow$ Only $\mathrm{S}_2$ correct ^

Asked in: JEE Main 2025 (08 Apr Shift 2)

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