Question

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 $ (S_1) : $ The number of elements in $ R $ is 18, and $ (S_2) : $ The relation $ R $ is symmetric but neither reflexive nor transitive Options 1. only $ (S_1) $ is true
2. both are true
3. only $ (S_2) $ is true
4. both are false

• A. only \( (S_1) \) is true
• B. both are true
• C. only \( (S_2) \) is true
• D. both are false

Hint:

- For relation counting, enumerate all valid pairs systematically - Check symmetry by verifying \((x,y) \in R \Rightarrow (y,x) \in R\) - Reflexivity requires \((a,a) \in R\) for all \(a \in A\) - Transitivity requires \((a,b), (b,c) \in R \Rightarrow (a,c) \in R\)

Answer 1

The Correct Option is C

To address this, we will evaluate the conditions for relation \( R \) on set \( A = \{0, 1, 2, 3, 4, 5\} \), where \( (x, y) \in R \) if and only if \(\max\{x, y\} \in \{3, 4\}\).

First, let's determine the pairs where \(\max\{x, y\} = 3\):

• If \( x = 3 \) or \( y = 3 \), then \(\max\{x, y\} = 3\). The applicable pairs are: \((3, 0), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (0, 3), (1, 3), (2, 3), (4, 3), (5, 3)\).

Next, identify the pairs where \(\max\{x, y\} = 4\):

• If \( x = 4 \) or \( y = 4 \), and \(\max\{x, y\} = 4\). The applicable pairs are: \((4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (0, 4), (1, 4), (2, 4), (3, 4), (5, 4)\).

Combining these, the complete set of ordered pairs in \( R \), without duplicates, is:

• \((3, 0), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (0, 3), (1, 3), (2, 3), (4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (0, 4), (1, 4), (2, 4), (5, 3), (3, 4)\)

The cardinality of \( R \) is \(20\), not 18. Consequently, Statement \( S_1 \) is false.

Next, we analyze whether relation \( R \) is symmetric, reflexive, or transitive:

• Symmetric: If \((x, y) \in R\), then \((y, x) \in R\). For every pair in \( R \), its inverse is also present. Therefore, the relation is symmetric.
• Reflexive: For all \( x \in A \), \((x, x) \in R\). This condition is not met, as pairs such as \( (0, 0), (1, 1), (2, 2) \) are absent from \( R \). Thus, the relation is not reflexive.
• Transitive: If \((x, y) \in R\) and \((y, z) \in R\), then \((x, z) \in R\). For instance, \((3, 0) \in R\) and \((0, 4) \in R\), but \((3, 4)\) is not in \( R \). Thus, the relation is not transitive.

Based on the preceding evaluation, Statement \( S_2 \), asserting that the relation is symmetric but neither reflexive nor transitive, is true.

Therefore, the correct conclusion is: only \( (S_2) \) is true.