Key Concepts and Formulas

• Relation: A relation $R$ on a set $A$ is a subset of $A \times A$. $(x, y) \in R$ means $x$ is related to $y$.
• Reflexive Relation: $\forall x \in A, (x, x) \in R$.
• Symmetric Relation: $\forall x, y \in A, (x, y) \in R \implies (y, x) \in R$.
• Transitive Relation: $\forall x, y, z \in A, [(x, y) \in R \land (y, z) \in R] \implies (x, z) \in R$.
• Counting pairs for $\max\{x,y\} = k$: For a set $S$, the number of pairs $(x,y) \in S \times S$ such that $\max\{x,y\} = k$ is $| \{x \in S \mid x \le k \}|^2 - | \{x \in S \mid x < k \}|^2$.

Step-by-Step Solution

Step 1: Determine the elements of the relation R and verify Statement (S1)

The set is $A = \{0, 1, 2, 3, 4, 5\}$. The relation $R$ is defined by $(x, y) \in R$ if and only if $\max\{x, y\} \in \{3, 4\}$. This means $\max\{x, y\} = 3$ or $\max\{x, y\} = 4$. We will consider these two cases separately.

Case 1: $\max\{x, y\} = 3$ For $\max\{x, y\} = 3$, we must have $x \le 3$, $y \le 3$, and at least one of $x$ or $y$ must be equal to 3. The elements in $A$ that are less than or equal to 3 are $\{0, 1, 2, 3\}$. There are 4 such elements. The elements in $A$ that are strictly less than 3 are $\{0, 1, 2\}$. There are 3 such elements. The number of pairs $(x, y)$ such that $\max\{x, y\} = 3$ is the number of pairs where $x, y \in \{0, 1, 2, 3\}$ minus the number of pairs where $x, y \in \{0, 1, 2\}$. Number of pairs = $(\text{number of elements } \le 3)^2 - (\text{number of elements } < 3)^2 = 4^2 - 3^2 = 16 - 9 = 7$. These pairs are: $(0,3), (1,3), (2,3), (3,3), (3,0), (3,1), (3,2)$.

Case 2: $\max\{x, y\} = 4$ For $\max\{x, y\} = 4$, we must have $x \le 4$, $y \le 4$, and at least one of $x$ or $y$ must be equal to 4. The elements in $A$ that are less than or equal to 4 are $\{0, 1, 2, 3, 4\}$. There are 5 such elements. The elements in $A$ that are strictly less than 4 are $\{0, 1, 2, 3\}$. There are 4 such elements. The number of pairs $(x, y)$ such that $\max\{x, y\} = 4$ is the number of pairs where $x, y \in \{0, 1, 2, 3, 4\}$ minus the number of pairs where $x, y \in \{0, 1, 2, 3\}$. Number of pairs = $(\text{number of elements } \le 4)^2 - (\text{number of elements } < 4)^2 = 5^2 - 4^2 = 25 - 16 = 9$. These pairs are: $(0,4), (1,4), (2,4), (3,4), (4,4), (4,0), (4,1), (4,2), (4,3)$.

The total number of elements in $R$ is the sum of the counts from these two mutually exclusive cases: $|R| = 7 (\text{for } \max=3) + 9 (\text{for } \max=4) = 16$.

Statement (S1) says the number of elements in $R$ is 18. Since $|R|=16$, Statement (S1) is False.

Step 2: Analyze the properties of R and verify Statement (S2)

Statement (S2) claims that the relation R is symmetric but neither reflexive nor transitive. We will check each property.

2.1. Reflexivity A relation $R$ is reflexive if $(x, x) \in R$ for all $x \in A$. This means $\max\{x, x\} \in \{3, 4\}$ for all $x \in \{0, 1, 2, 3, 4, 5\}$. $\max\{x, x\} = x$. So, we need $x \in \{3, 4\}$ for all $x \in A$. Let's test for $x=0$: $\max\{0, 0\} = 0$. Since $0 \notin \{3, 4\}$, $(0, 0) \notin R$. Since $(0, 0) \notin R$, the relation $R$ is not reflexive. This part of (S2) is true.

2.2. Symmetry A relation $R$ is symmetric if $(x, y) \in R \implies (y, x) \in R$. If $(x, y) \in R$, then $\max\{x, y\} \in \{3, 4\}$. Since $\max\{y, x\} = \max\{x, y\}$, it follows that if $\max\{x, y\} \in \{3, 4\}$, then $\max\{y, x\} \in \{3, 4\}$. Thus, if $(x, y) \in R$, then $(y, x) \in R$. The relation $R$ is symmetric. This part of (S2) is true.

2.3. Transitivity A relation $R$ is transitive if $(x, y) \in R$ and $(y, z) \in R$ implies $(x, z) \in R$. Let's check for a counterexample. Consider the pair $(0, 3)$. $\max\{0, 3\} = 3$, so $(0, 3) \in R$. Consider the pair $(3, 0)$. $\max\{3, 0\} = 3$, so $(3, 0) \in R$. Now we must check if $(0, 0) \in R$. For $(0, 0)$ to be in $R$, we need $\max\{0, 0\} \in \{3, 4\}$. $\max\{0, 0\} = 0$. Since $0 \notin \{3, 4\}$, $(0, 0) \notin R$. We have found a case where $(0, 3) \in R$ and $(3, 0) \in R$, but $(0, 0) \notin R$. Therefore, the relation $R$ is not transitive. This part of (S2) is true.

Since $R$ is symmetric, not reflexive, and not transitive, Statement (S2) "The relation R is symmetric but neither reflexive nor transitive" is True.

Step 3: Evaluate the options based on the truth values of (S1) and (S2)

• Statement (S1): False
• Statement (S2): True

We need to find the option that states both are false.

Common Mistakes & Tips

• When counting elements for $\max\{x, y\} = k$, always ensure you are considering pairs from the correct subset of $A$. The formula $(|S_{\le k}|)^2 - (|S_{< k}|)^2$ is very useful for such counting.
• For reflexivity, a single element $x \in A$ for which $(x, x) \notin R$ is sufficient to prove it's not reflexive.
• Transitivity often requires careful searching for counterexamples. Look for chains like $(a,b) \in R$ and $(b,c) \in R$ where $a, b, c$ might be related in a way that breaks the transitive property.

Summary

We first determined the number of elements in the relation $R$ by considering the conditions $\max\{x, y\} = 3$ and $\max\{x, y\} = 4$. This calculation showed that $|R| = 16$, making statement (S1) false. Subsequently, we analyzed the properties of $R$: it is not reflexive because $(0,0) \notin R$; it is symmetric because $\max\{x,y\} = \max\{y,x\}$; and it is not transitive, as demonstrated by the counterexample $(0,3) \in R$ and $(3,0) \in R$ but $(0,0) \notin R$. Therefore, statement (S2) is true. With (S1) being false and (S2) being true, both statements are false is the correct conclusion.

The final answer is \boxed{A}.