JEE Main 2025 — Sets And Relations Question with Solution

To evaluate the relation $R$ on the set $A=\{0,1,2,3,4,5\}$, we first need to understand the conditions for an element $(x,y)$ to be in $R$. Specifically, $(x,y)\in R$ if and only if $\max \{x,y\}\in \{3,4\}$.

Considering this, let's list the pairs:

For $\max \{x,y\}=3$, the possible pairs are:

$(0,3),(3,0),(1,3),(3,1),(2,3),(3,2),(3,3)$

For $\max \{x,y\}=4$, the possible pairs are:

$(0,4),(4,0),(1,4),(4,1),(2,4),(4,2),(3,4),(4,3),(4,4)$

Combining these, the set $R$ consists of the following elements:

$R=\{(0,3),(3,0),(1,3),(3,1),(2,3),(3,2),(3,3),(0,4),(4,0),(1,4),(4,1),(2,4),(4,2),(3,4),(4,3),(4,4)\}$

This gives us a total of 16 elements in $R$, not 18 as initially claimed in statement $S_1$.

Next, we analyze the properties of the relation $R$:

Reflexivity: A relation is reflexive if $(x,x)\in R$ for all $x\in A$. For example, $(0,0),(1,1),(2,2)$ are not in $R$, so $R$ is not reflexive.

Symmetry: A relation is symmetric if whenever $(a,b)\in R$, then $(b,a)\in R$ as well. For all pairs $(x,y)$ listed, both $(x,y)$ and $(y,x)$ are present. Thus, $R$ is symmetric.

Transitivity: A relation is transitive if whenever $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$. An example where transitivity fails is $(0,3)$ and $(3,1)$ are in $R$ but $(0,1)$ is not in $R$. Therefore, $R$ is not transitive.

In conclusion, statement $S_2$ is correct as $R$ is symmetric but neither reflexive nor transitive.