Question

Let \( A = (1, 2, 3, \dots, 20) \). Let \( R \subseteq A \times A \) such that \( R = \{(x, y) : y = 2x - 7 \ \). Then the number of elements in \( R \) is equal to:}

• A. 7
• B. 13
• C. 17
• D. 10

Hint:

To find the number of solutions for an equation, first determine the valid range for the variable and then count the possible integer values.

Answer 1

The Correct Option is D

We are given that \( R \) is the set of ordered pairs \( (x, y) \) such that \( y = 2x - 7 \), and \( x \) and \( y \) belong to the set \( A = \{1, 2, 3, \dots, 20\} \). For each \( x \in A \), we can compute \( y = 2x - 7 \). To ensure that \( y \) is in \( A \), it must satisfy: \[ 1 \leq y \leq 20 \] Substituting \( y = 2x - 7 \), we get: \[ 1 \leq 2x - 7 \leq 20 \] Solving this inequality: \[ 1 + 7 \leq 2x \leq 20 + 7 \] \[ 8 \leq 2x \leq 27 \] \[ 4 \leq x \leq 13.5 \] Since \( x \) must be an integer, the possible values of \( x \) are \( x = 4, 5, 6, \dots, 13 \). Thus, there are 10 values for \( x \), so the number of elements in \( R \) is \( 10 \).