Question

Let $A = \\{1, 2, 3, \ldots, 100\\}$. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $2x = 3y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $n$. Then, the minimum value of $n$ is \\_\\_\\_\\_\\_.

Answer 1

The relation $R$ consists of ordered pairs $(x, y)$ such that $2x = 3y$. For $x$ and $y$ to satisfy this relation, $x$ and $y$ must form pairs with specific integer values that satisfy $2x = 3y$.

Thus, the pairs in $R$ are:

$R = \\{(3, 2), (6, 4), (9, 6), (12, 8), \ldots, (99, 66)\\}.$

There are 33 such pairs in $R$, so:

$n(R) = 33.$

To make $R_1$ symmetric, we include both $(x, y)$ and $(y, x)$ for each pair in $R$. Thus, the pairs in $R_1$ are:

$R_1 = \\{(3, 2), (2, 3), (6, 4), (4, 6), (9, 6), (6, 9), \ldots, (99, 66), (66, 99)\\}.$

This doubles the number of elements:

$n = 2 \times 33 = 66.$

Therefore, the minimum value of $n$ is: $66$