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

Given: \( 4x \leq 5y \)

then

\[ R = \{(1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,4), (5,5)\} \]

i.e., 16 elements.

i.e., \( n = 16 \)

Now to make \( R \) a symmetric relation, add:

\[ \{(2,1), (3,2), (4,3), (1,4), (2,5), (3,4), (1,5), (2,1)\} \]

i.e., \( m = 9 \)

So \( m + n = 25 \)