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

We are given the set \( A = \{0, 1, 2, 3, 4, 5, 6\} \) and the relation \( R_1 \) defined by \( \max(x, y) \in \{3, 4\} \), meaning \( R_1 \) consists of all 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 \( \max(x, y) = 3 \), \( x \) and \( y \) can take values from the set \( \{0, 1, 2, 3\} \), giving \( 4 \times 4 = 16 \) pairs.
For \( \max(x, y) = 4 \), \( x \) and \( y \) can take values from the set \( \{0, 1, 2, 3, 4\} \), giving \( 5 \times 5 = 25 \) pairs.
Thus, the total number of elements in \( R_1 \) is: \[ 16 + 25 = 41 \] 
Therefore, Statement 1 is false because the total number of elements is 41, not 18. 
Statement 2: Symmetry, Reflexivity, and Transitivity
Symmetry: A relation \( R \) is symmetric if for all \( (x, y) \in R \), \( (y, x) \in R \).
Since \( \max(x, y) \) is symmetric (i.e., if \( \max(x, y) = 3 \), then \( \max(y, x) = 3 \), and similarly for 4), the relation \( R_1 \) is symmetric. Reflexivity: A relation \( R \) is reflexive if for every element \( x \in A \), \( (x, x) \in R \).
For \( R_1 \), \( \max(x, x) = x \), but \( (x, x) \) will not satisfy the condition for values of \( x \) other than 3 and 4.
Hence, \( R_1 \) is not reflexive. Transitivity: A relation \( R \) is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).
\( R_1 \) is not transitive because the relation depends only on the maximum of the values and does not maintain the transitive property. Thus, Statement 2 is true, as \( R_1 \) is symmetric but neither reflexive nor transitive.