Question

Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). A relation \( R \) is defined such that \( xRy \) iff \( y = \max(x, 1) \). The number of elements required to make it reflexive is \( l \), the number of elements required to make it symmetric is \( m \), and the number of elements in the relation \( R \) is \( n \). Then the value of \( l + m + n \) is equal to:

• A. 7
• B. 8
• C. 9
• D. 10

Hint:

To make a relation reflexive, ensure that every element is related to itself. To make it symmetric, ensure that for every \( (x, y) \), the pair \( (y, x) \) is also included.

Answer 1

The Correct Option is C

Given that the relation \( xRy \) is defined by \( y = \max(x, 1) \), we can understand that for every element in set \( A \), the relation holds if the value of \( y \) is the greater of \( x \) and 1. 1. Reflexive: A relation is reflexive if every element is related to itself. For reflexivity, the relation must include \( (x, x) \) for every \( x \in A \). In this case, \( xRx \) will hold only if \( x \geq 1 \). Therefore, the elements that need to be added to make the relation reflexive are those less than 1, which are \( -3, -2, -1, 0 \). Hence, the number of elements required to make the relation reflexive is \( l = 4 \). 2. Symmetric: A relation is symmetric if for every \( (x, y) \in R \), we must also have \( (y, x) \in R \). Here, the relation is asymmetric for elements where \( x \neq y \), but it would become symmetric if we add \( (1, 2) \) and \( (2, 1) \) to the relation. Hence, the number of elements required to make the relation symmetric is \( m = 2 \). 3. Number of elements in the relation: The number of elements in the relation is determined by the number of ordered pairs that satisfy \( y = \max(x, 1) \). Since \( A \) has 7 elements, the total number of elements in the relation is \( n = 7 \). Thus, the value of \( l + m + n \) is: \[ l + m + n = 4 + 2 + 7 = 9 \]