Question

Relation \( R \) on the set \( A = \{1, 2, 3, \ldots, 13, 14\} \) defined as \( R = \{(x, y) : 3x - y = 0\} \) is:

• A. Reflexive, symmetric and transitive
• B. Reflexive and transitive but not symmetric
• C. Neither reflexive nor symmetric but transitive
• D. Neither reflexive nor symmetric nor transitive

Answer 1

The Correct Option is C

To establish the nature of the relation \( R \) on the set \( A = \{1, 2, 3, \ldots, 14\} \), defined by \( R = \{(x, y) : 3x - y = 0\} \), we examine its properties: reflexivity, symmetry, and transitivity.

1. Reflexivity: A relation is reflexive if \( (x, x) \in R \) for all \( x \in A \). For \( R \), this would require \( 3x - x = 0 \), which simplifies to \( 2x = 0 \). This condition is only met if \( x = 0 \), but \( 0 otin A \). Therefore, \( R \) is not reflexive.

2. Symmetry: A relation is symmetric if \( (x, y) \in R \) implies \( (y, x) \in R \). If \( (x, y) \in R \), then \( 3x - y = 0 \), so \( y = 3x \). For symmetry, we would also need \( (y, x) \in R \), meaning \( 3y - x = 0 \). Substituting \( y = 3x \) into this second equation gives \( 3(3x) - x = 0 \), which simplifies to \( 9x - x = 0 \), or \( 8x = 0 \). This implies \( x = 0 \), which is not in \( A \). Thus, \( R \) is not symmetric.

3. Transitivity: A relation is transitive if \( (x, y) \in R \) and \( (y, z) \in R \) imply \( (x, z) \in R \). Given \( (x, y) \in R \), we have \( y = 3x \). Given \( (y, z) \in R \), we have \( z = 3y \). Substituting the first equation into the second gives \( z = 3(3x) = 9x \). For \( (x, z) \) to be in \( R \), we would need \( 3x - z = 0 \). Substituting \( z = 9x \) into this gives \( 3x - 9x = 0 \), which simplifies to \( -6x = 0 \), implying \( x = 0 \). However, the definition of transitivity requires that if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \) must hold for all valid \( x, y, z \in A \). Consider \( x=1 \). Then \( y=3(1)=3 \), so \( (1,3) \in R \). Since \( 3 \in A \) and \( y=3 \), we check for \( z \). If \( (3, z) \in R \), then \( z = 3(3) = 9 \). So \( (3,9) \in R \). For transitivity, we require \( (1,9) \in R \). Checking this, \( 3(1) - 9 = 3 - 9 = -6 eq 0 \). Therefore, \( (1,9) otin R \), and \( R \) is not transitive.

Therefore, the correct answer is: Neither reflexive, nor symmetric, nor transitive.