Let A be any set. There exist two equivalence relations ≡coarsest and ≡finest with the following property: if ≡ is an equivalence relation on A, then (i) ≡ refines ≡coarsest, and (ii) ≡finest refines...

Let A be any set. There exist two equivalence relations ≡coarsest and ≡finest with the following property: if ≡ is an equivalence relation on A, then (i) ≡ refines ≡coarsest, and (ii) ≡finest refines ≡.

1. Identify ≡coarsest, prove that it’s an equivalence relation, and prove property (i) above.

2. Identify ≡finest, prove that it’s an equivalence relation, and prove property (ii) above.