Question

Let $A = \{1, 2, 3, 4, 5\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $4x \leq 5y$. Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m + n$ is equal to:

• A. 24
• B. 23
• C. 25
• D. 26

Answer 1

The Correct Option is C

To address this problem, we will analyze the relation \( R \) defined on the set \( A = \{1, 2, 3, 4, 5\} \). The condition for \( xRy \) is \( 4x \leq 5y \).

Initially, we identify and list all pairs \((x, y)\) that satisfy the condition \( 4x \leq 5y \):

\( x \)	Possible \( y \)
1	\( (1, 1), (1, 2), (1, 3), (1, 4), (1, 5) \)
2	\( (2, 2), (2, 3), (2, 4), (2, 5) \)
3	\( (3, 3), (3, 4), (3, 5) \)
4	\( (4, 4), (4, 5) \)
5	\( (5, 5) \)

The total count of these initial pairs in \( R \) is \( m = 15 \).

Next, we aim to make \( R \) symmetric. A relation is symmetric if for every pair \((x, y) \in R\), the pair \((y, x)\) is also in \( R \). Our objective is to achieve symmetry by adding the fewest possible pairs.

The analysis for symmetry is as follows:

• For \( x = 1 \): All pairs \((1, y)\) are covered because their corresponding \((y, 1)\) pairs are either already present or are identical to the original pairs.
• For \( x = 2 \): The pairs \((2, 3)\), \((2, 4)\), \((2, 5)\) require the addition of \((3, 2)\), \((4, 2)\), \((5, 2)\) to ensure symmetry.
• For \( x = 3 \): The pairs \((3, 4)\) and \((3, 5)\) necessitate the addition of \((4, 3)\) and \((5, 3)\).
• For \( x = 4 \): The pair \((4, 5)\) requires the addition of \((5, 4)\).

The additional pairs required for symmetry are \((3, 2), (4, 2), (5, 2), (4, 3), (5, 3), (5, 4)\). Therefore, we need to add \( n = 6 \) pairs. This calculation is derived from (2,3) → (3,2), (2,4) → (4,2), (2,5) → (5,2), (3,4) → (4,3), (3,5) → (5,3), (4,5) → (5,4). Note that for \( x = 1 \), all \( y \) are covered. For \( x = 2 \), we need to add \((3, 2)\), \((4, 2)\), \((5, 2)\). For \( x = 3 \), we need to add \((4, 3)\), \((5, 3)\). For \( x = 4 \), we need to add \((5, 4)\). For \( x = 5 \), no new pairs are needed as \((5, 5)\) is symmetric. Thus \( n = 6 \) pairs, not 10.

Upon re-evaluating the pairs needing symmetry:

• For \( (2, 3) \), we need \( (3, 2) \).
• For \( (2, 4) \), we need \( (4, 2) \).
• For \( (2, 5) \), we need \( (5, 2) \).
• For \( (3, 4) \), we need \( (4, 3) \).
• For \( (3, 5) \), we need \( (5, 3) \).
• For \( (4, 5) \), we need \( (5, 4) \).

This totals \( n = 6 \) additional pairs.

The total number of pairs after adding the necessary ones for symmetry is \( m + n = 15 + 6 = 21 \).

The final answer is 21, representing:

• The initial relation contains 15 pairs satisfying \( 4x \leq 5y \).
• 6 additional pairs were added to ensure the relation is symmetric.