(Solved): Detailed written proof in sentences. Thank you. 1. Define a relation \( \mathrm{R} \) on the set of ...

Detailed written proof in sentences. Thank you.

1. Define a relation \( \mathrm{R} \) on the set of \( S=\mathbb{R} \times \mathbb{R} \) by \( (a, b) \mathrm{R}(c, d) \) iff \( a+2 b=c+2 d \). (a) Prove that \( \mathrm{R} \) is an equivalence relation on \( S \) (b) Describe the elements of \( S \) in the equivalence class \( E_{(5,3)} \)