Section 1.4: Composition of Functions

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, \(R_2 \subset R_1 \Rightarrow R_1 = R_2\).