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
We are given the set $A = \{0, 1, 2, 3, 4, 5\}$ and a relation $R$ defined on $A$ such that $(x, y) \in R$ if and only if $\max \{x, y\} \in \{3, 4\}$.
Analyzing Statement (S₁): Number of elements in R is 18
For a pair $(x, y)$ to be in $R$, we must have $\max \{x, y\} = 3$ or $\max \{x, y\} = 4$. This implies that both $x$ and $y$ must be less than or equal to 4. Therefore, elements equal to 5 in set $A$ cannot be part of any pair in $R$. We only need to consider the subset $A' = \{0, 1, 2, 3, 4\}$.
Case 1: $\max \{x, y\} = 3$. The pairs $(x, y)$ from $A'$ are:
- If $x=3$: $(3, 0), (3, 1), (3, 2), (3, 3)$ (4 pairs)
- If $y=3$ and $x \ne 3$: $(0, 3), (1, 3), (2, 3)$ (3 pairs)
- Total pairs where $\max \{x, y\} = 3$: $4 + 3 = 7$.
Case 2: $\max \{x, y\} = 4$. The pairs $(x, y)$ from $A'$ are:
- If $x=4$: $(4, 0), (4, 1), (4, 2), (4, 3), (4, 4)$ (5 pairs)
- If $y=4$ and $x \ne 4$: $(0, 4), (1, 4), (2, 4), (3, 4)$ (4 pairs)
- Total pairs where $\max \{x, y\} = 4$: $5 + 4 = 9$.
The total number of elements in $R$ is the sum of pairs from Case 1 and Case 2: $7 + 9 = 16$. Statement (S₁) claims the number of elements is 18, which is incorrect. Therefore, statement (S₁) is False.
Analyzing Statement (S₂): R is symmetric, not reflexive, not transitive
1. Reflexivity: A relation $R$ is reflexive if $(x, x) \in R$ for all $x \in A$. Let's check $x=0$. We need $\max \{0, 0\} \in \{3, 4\}$. Since $\max \{0, 0\} = 0$, which is not in $\{3, 4\}$, $(0, 0) \notin R$. Thus, $R$ is not reflexive.
2. Symmetry: A relation $R$ is symmetric if whenever $(x, y) \in R$, then $(y, x) \in R$. If $(x, y) \in R$, then $\max \{x, y\} \in \{3, 4\}$. Since $\max \{y, x\} = \max \{x, y\}$, it follows that $\max \{y, x\} \in \{3, 4\}$. Therefore, $(y, x) \in R$. Thus, $R$ is symmetric.
3. Transitivity: A relation $R$ is transitive if whenever $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$. Consider the pairs $(0, 3)$ and $(3, 0)$. We have $\max \{0, 3\} = 3$, so $3 \in \{3, 4\}$, hence $(0, 3) \in R$. We have $\max \{3, 0\} = 3$, so $3 \in \{3, 4\}$, hence $(3, 0) \in R$. Now consider the pair $(x, z) = (0, 0)$. We need to check if $(0, 0) \in R$. $\max \{0, 0\} = 0$. Since $0 \notin \{3, 4\}$, $(0, 0) \notin R$. Since we found a case where $(x, y) \in R$ and $(y, z) \in R$ but $(x, z) \notin R$, the relation $R$ is not transitive.
Combining the properties: $R$ is symmetric, not reflexive, and not transitive. Therefore, statement (S₂) is True.
Conclusion
Statement (S₁) is False and Statement (S₂) is True. Thus, only (S₂) is true.
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