Question

Let $ A = \{0, 1, 2, 3, 4, 5, 6\} $ and $ R_1 = \{(x, y): \max(x, y) \in \{3, 4 \}$. Consider the two statements:
Statement 1: Total number of elements in $ R_1 $ is 18.
Statement 2: $ R $ is symmetric but not reflexive and transitive.

• A. Statement 1 is true, but Statement 2 is false
• B. Statement 2 is true, but Statement 1 is false
• C. Statement 1 and Statement 2 are true
• D. Neither Statement 1 nor Statement 2 are true

Hint:

When determining the properties of relations, check the definitions of symmetry, reflexivity, and transitivity, and evaluate each condition individually.

Answer 1

The Correct Option is B

Given the set \( A = \{0, 1, 2, 3, 4, 5, 6\} \) and relation \( R_1 \) defined by \( \max(x, y) \in \{3, 4\} \). This means \( R_1 \) contains pairs \( (x, y) \) where the maximum of \( x \) and \( y \) is either 3 or 4.
Statement 1: Total number of elements in \( R_1 \)
For \( \max(x, y) = 3 \), \( x, y \in \{0, 1, 2, 3\} \). This yields \( 4 \times 4 = 16 \) pairs.
For \( \max(x, y) = 4 \), \( x, y \in \{0, 1, 2, 3, 4\} \). This yields \( 5 \times 5 = 25 \) pairs.
The total number of elements in \( R_1 \) is \( 16 + 25 = 41 \).
Therefore, Statement 1 is false as the calculated total is 41, not 18.
Statement 2: Symmetry, Reflexivity, and Transitivity
Symmetry: \( R \) is symmetric if \( (x, y) \in R \Rightarrow (y, x) \in R \). Since \( \max(x, y) = \max(y, x) \), \( R_1 \) is symmetric.
Reflexivity: \( R \) is reflexive if \( x \in A \Rightarrow (x, x) \in R \). For \( R_1 \), \( \max(x, x) = x \). This condition is met only for \( x=3 \) and \( x=4 \). Therefore, \( R_1 \) is not reflexive.
Transitivity: \( R \) is transitive if \( (x, y) \in R \) and \( (y, z) \in R \Rightarrow (x, z) \in R \). \( R_1 \) is not transitive as it only considers the maximum value and does not preserve the transitive property.
Thus, Statement 2 is true, as \( R_1 \) is symmetric but neither reflexive nor transitive.