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 (S₁): The number of elements in R is 18, and (S₂): The relation R is symmetric but neither reflexive nor transitive Which of the following is correct?

Step 1: Determine the elements of the relation R

The relation $R$ is defined on the set $A = \{0, 1, 2, 3, 4, 5\}$. A pair $(x, y)$ belongs to $R$ if the maximum of $x$ and $y$ is either 3 or 4. We need to list all such pairs.

Step 2: List pairs where max(x, y) = 3

If $\max\{x, y\} = 3$, then both $x$ and $y$ must be less than or equal to 3, and at least one of them must be 3. The possible pairs are $(0{,}3), (1{,}3), (2{,}3), (3{,}3), (3{,}0), (3{,}1), (3{,}2)$. There are 7 such pairs.

Step 3: List pairs where max(x, y) = 4

If $\max\{x, y\} = 4$, then both $x$ and $y$ must be less than or equal to 4, and at least one of them must be 4. The possible pairs are $(0{,}4), (1{,}4), (2{,}4), (3{,}4), (4{,}4), (4{,}0), (4{,}1), (4{,}2), (4{,}3)$. There are 9 such pairs.

Step 4: Calculate the total number of elements in R

The total number of elements in $R$ is the sum of the pairs where $\max\{x, y\} = 3$ and the pairs where $\max\{x, y\} = 4$. So, $|R| = 7 + 9 = 16$. Therefore, statement (S $_{1}$) which claims $|R|=18$ is incorrect.

Step 5: Check properties of the relation R

1. Reflexivity: For $R$ to be reflexive, $(a,a)$ must be in $R$ for all $a \in A$. However, for $a=0$, $\max\{0{,}0\} = 0 \notin \{3{,}4\}$. So $(0{,}0) \notin R$, which means $R$ is not reflexive. 2. Symmetry: If $(x,y) \in R$, then $\max\{x,y\} \in \{3{,}4\}$. Since $\max\{x,y\} = \max\{y,x\}$, it follows that $\max\{y,x\} \in \{3{,}4\}$, which means $(y,x) \in R$. Thus, $R$ is symmetric. 3. Transitivity: For $R$ to be transitive, if $(x,y) \in R$ and $(y,z) \in R$, then $(x,z) \in R$. Consider $(2{,}3) \in R$ (since $\max\{2{,}3\}=3$) and $(3{,}0) \in R$ (since $\max\{3{,}0\}=3$). However, $(2{,}0) \notin R$ because $\max\{2{,}0\}=2 \notin \{3{,}4\}$. Therefore, $R$ is not transitive. Statement (S $_{2}$) is correct.

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