Question

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to :

• A. 23
• B. 21
• C. 25
• D. 27

Hint:

For symmetry, ignore diagonal elements (where $x=y$) and check if every pair $(x,y)$ has its reflection $(y,x)$ already in the set.

Answer 1

The Correct Option is C

Given the set \[ A = \{2, 3, 5, 7, 9\} \] and a relation \( R \) defined by \[ xRy \iff 2x \le 3y, \] we are required to:

• Find the number of elements in \( R \).
• Determine how many more ordered pairs are needed to make \( R \) symmetric.

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Step 1: Total Possible Ordered Pairs

Since \( |A| = 5 \), the total number of ordered pairs in \( A \times A \) is:

\[ 5 \times 5 = 25 \]

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Step 2: Evaluate the Condition \( 2x \le 3y \)

Checking each ordered pair:

• (2,2), (2,3), (2,5), (2,7), (2,9) ✔
• (3,2), (3,3), (3,5), (3,7), (3,9) ✔
• (5,5), (5,7), (5,9) ✔
• (7,5), (7,7), (7,9) ✔
• (9,7), (9,9) ✔

All remaining pairs violate the inequality.

\[ |R| = 17 \]

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Step 3: Making the Relation Symmetric

A relation is symmetric if:

\[ (x,y) \in R \Rightarrow (y,x) \in R \]

Check which ordered pairs in \( R \) do not have their reverse in \( R \). These missing symmetric counterparts are:

• (5,2)
• (7,2)
• (7,3)
• (9,2)
• (9,3)
• (9,5)
• (5,3)
• (7,3)

Number of missing pairs:

\[ m = 8 \]

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Step 4: Total Elements After Symmetrization

\[ \text{Total} = 17 + 8 = 25 \]

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Final Answer:

\(\boxed{25}\)