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 solve this problem, we will examine the conditions for the relation \( R \) defined 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 identify the pairs where \(\max\{x, y\} = 3\):

• If \( x = 3 \) or \( y = 3 \), then \(\max\{x, y\} = 3\). The possible 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 possible 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 repetition, 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 number of elements in \( R \) is \(20\), not 18. Therefore, Statement \( S_1 \) is false.

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

• Symmetric: If \((x, y) \in R\), then \((y, x) \in R\). For each pair in \( R \), its reverse is also in \( R \). Thus, the relation is symmetric.
• Reflexive: For all \( x \in A \), \((x, x) \in R\). This fails as elements such as \( (0, 0), (1, 1), (2, 2) \) are not in \( R \). Thus, the relation is not reflexive.
• Transitive: If \((x, y) \in R\) and \((y, z) \in R\), then \((x, z) \in R\). For example, \((3, 0) \in R\) and \((0, 4) \in R\) but \((3, 4)\) is not in \( R\). Thus, the relation is not transitive.

Based on the above analysis, Statement \( S_2 \), which states that the relation is symmetric but neither reflexive nor transitive, is true.

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

Answer 2

Analysis of Relation \( R \): 
1. Definition: \((x, y) \in R\) iff \(\max\{x, y\} \in \{3, 4\}\)
2. Counting elements in \( R \):
- Pairs where \(\max = 3\): \((0,3), (1,3), (2,3), (3,0), (3,1), (3,2), (3,3)\)
- Pairs where \(\max = 4\): \((0,4), (1,4), (2,4), (3,4), (4,0), (4,1), (4,2), (4,3), (4,4)\)
- Total pairs = 7 (for 3) + 9 (for 4) = 16
- Thus, \( (S_1) \) is false (claims 18)
3. Properties of \( R \):
- Symmetric: If \((x,y) \in R\), then \((y,x) \in R\) since \(\max\) is symmetric
- Not Reflexive: \((5,5) \notin R\) since \(\max\{5,5\} = 5 \notin \{3,4\}\)
- Not Transitive: Counterexample: \((0,3) \in R\) and \((3,4) \in R\), but \((0,4) \notin R\)
- Thus, \( (S_2) \) is true