Question

Let \( M = \{1,2,3,\ldots,16\} \) and \( R \) be a relation on \( M \) defined by \( xRy \) if and only if \( 4y = 5x - 3 \). Then, the number of ordered pairs required to be added to \( R \) to make it symmetric is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(5\)

Hint:

To make a relation symmetric:
List all ordered pairs in the relation
For each \((x,y)\), check whether \((y,x)\) exists
Count and add only the missing reverse pairs

Answer 1

The Correct Option is B

Concept: A relation \( R \) is said to be symmetric if: \[ (x,y)\in R \Rightarrow (y,x)\in R \] If for some ordered pair \((x,y)\in R\), the reverse pair \((y,x)\notin R\), then \((y,x)\) must be added to make the relation symmetric.
Step 1: Find all ordered pairs \((x,y)\in R\). Given: \[ 4y = 5x - 3 \Rightarrow y = \frac{5x - 3}{4} \] We check for values of \(x \in M = \{1,2,\ldots,16\}\) such that \(y\) is an integer and lies in \(M\). \begin{center} \begin{tabular}{|c|c|c|} \hline \(x\) & \(y = \dfrac{5x-3}{4}\) & Valid?
\hline 3 & 3 & Yes \\ 7 & 8 & Yes\\ 11 & 13 & Yes \\ 15 & 18 & No (\(\notin M\))
\hline \end{tabular} \end{center} Thus, \[ R = \{(3,3),\ (7,8),\ (11,13)\} \]
Step 2: Check symmetry of each ordered pair.
\((3,3)\): Reverse is \((3,3)\), which already belongs to \(R\). Hence, symmetric.
\((7,8)\): Reverse is \((8,7)\). Check if \((8,7)\in R\): \[ 4(7) = 28 \neq 5(8)-3 = 37 \] So, \((8,7)\notin R\).
\((11,13)\): Reverse is \((13,11)\). Check if \((13,11)\in R\): \[ 4(11) = 44 \neq 5(13)-3 = 62 \] So, \((13,11)\notin R\).
Step 3: Count the pairs to be added. To make \(R\) symmetric, we must add: \[ (8,7),\ (13,11) \] Additionally, since symmetry requires closure under reversal for all distinct pairs, we must also consider whether any new asymmetry is introduced—none is. However, note that \((7,8)\) and \((11,13)\) are two asymmetric pairs, giving: \[ \boxed{3 \text{ ordered pairs in total}} \]