1. Define a relation R on the set of all integers Z by x R y iff x – y = 3k for some integer k. Verify that R is an equivalence relation and describe the equivalence class E 5 . How many distinct...

1. Define a relation R on the set of all integers Z by x R y iff x – y = 3k for some integer k. Verify that R is an equivalence relation and describe the equivalence class E₅. How many distinct equivalence classes are there?

2. Define a relation R on the set of all integers Z by x R y iff x + y = 2k for some integer k. Is R an equivalence relation on Z? Why or why not? (Compare with Exercise 23.)

Exercise 23

Define a relation R on the set of all integers Z by x R y iff x – y = 2k for some integer k. Verify that R is an equivalence relation and describe the equivalence class E5. How many distinct equivalence classes are there?