Let \(X = \{1, 2, 3, 4, 5, 6, 7, 8,
9\}\).
Let \(R_1\) be a relation in \(X\) given by \(R_1 = \{(x, y) : x - y\)</span> is divisible by
<span class="math inline">\(3\}\) and \(R_2\) be another relation on \(X\) given by \(R_2 = \{(x, y) : \{x, y\} \subset \{1, 4,
7\}\) or \(\{x, y\} \subset \{2, 5,
8\}\) or \(\{x, y\} \subset \{3, 6,
9\}\}\).
Show that \(R_1 = R_2\).
Solution
Note that the characteristic of sets \(\{1, 4,
7\}\), \(\{2, 5, 8\}\) and \(\{3, 6, 9\}\) is that difference between
any two elements of these sets is a multiple of 3.
Therefore, \((x, y) \in R_1 \Rightarrow x -
y\) is a multiple of 3 \(\Rightarrow
\{x, y\} \subset \{1, 4, 7\}\) or \(\{x, y\} \subset \{2, 5, 8\}\) or \(\{x, y\} \subset \{3, 6, 9\} \Rightarrow (x, y)
\in R_2\).
Hence, \(R_1 \subset R_2\).
Similarly, \((x, y) \in R_2 \Rightarrow x -
y\) is divisible by 3 \(\Rightarrow (x,
y) \in R_1\).
Thus, \(R_2 \subset R_1 \Rightarrow R_1 =
R_2\).