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

Answer: (C)

25

Answer 2

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 $m + n = 25$