QPaperGen

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 \(\max\{x, y\} \in \{3, 4\}\).


\[\] Number of elements in R (\(S_1\)):

For \(\max\{x, y\} = 3\), the possible pairs are \((0, 3), (3, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3)\). There are 7 pairs.\[\]
For \(\max\{x, y\} = 4\), the possible pairs are \((0, 4), (4, 0), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)\). There are 9 pairs.

\[\] Combining these, the set \(R\) consists of a total of \(7 + 9 = 16\) elements. Therefore, statement \((S_1)\) is false.

\[\] Properties of R (\(S_2\)):

\[\] Reflexivity: A relation is reflexive if \((x, x) \in R\) for all \(x \in A\). For a pair \((x, x)\) to be in \(R\), \(\max\{x, x\} = x\) must be in \(\{3, 4\}\). This is not true for all elements in \(A\) (e.g., \((0, 0)\) is not in \(R\)). So, \(R\) is not reflexive.
\[\] Symmetry: A relation is symmetric if whenever \((x, y) \in R\), then \((y, x) \in R\). Since \(\max\{x, y\} = \max\{y, x\}\), if \((x, y) \in R\), then \((y, x)\) is also in \(R\). So, \(R\) is symmetric.
\[\] Transitivity: A relation is transitive if whenever \((x, y) \in R\) and \((y, z) \in R\), then \((x, z) \in R\). For example, \((0, 3) \in R\) and \((3, 1) \in R\), but \((0, 1) \notin R\) because \(\max\{0, 1\} = 1 \notin \{3, 4\}\). So, \(R\) is not transitive.

\[\] Statement \((S_2)\) claims that \(R\) is symmetric but neither reflexive nor transitive, which is correct. Therefore, statement \((S_2)\) is true.