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

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