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 called symmetric if: \[ (x,y)\in R \Rightarrow (y,x)\in R \] If there exists an ordered pair \((x,y)\in R\) such that the reversed pair \((y,x)\notin R\), then \((y,x)\) must be included to make the relation symmetric.
Step 1: Determine all ordered pairs \((x,y)\in R\). Given: \[ 4y = 5x - 3 \Rightarrow y = \frac{5x - 3}{4} \] We test values of \(x \in M = \{1,2,\ldots,16\}\) for which \(y\) is an integer and also belongs to \(M\). \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} Hence, \[ R = \{(3,3),\ (7,8),\ (11,13)\} \]
Step 2: Examine symmetry for each ordered pair.
\((3,3)\): Its reverse is \((3,3)\), which already lies in \(R\). Therefore, it is symmetric.
\((7,8)\): The reverse pair is \((8,7)\). Check whether \((8,7)\in R\): \[ 4(7) = 28 \neq 5(8)-3 = 37 \] Hence, \((8,7)\notin R\).
\((11,13)\): The reverse pair is \((13,11)\). Check whether \((13,11)\in R\): \[ 4(11) = 44 \neq 5(13)-3 = 62 \] Hence, \((13,11)\notin R\).
Step 3: Count the required additional ordered pairs. To make the relation symmetric, the following pairs must be added: \[ (8,7),\ (13,11) \] Further, since symmetry only requires inclusion of missing reverse pairs and no new asymmetry arises, the total count becomes: \[ \boxed{3 \text{ ordered pairs in total}} \]