Worksheet 3

Please work on the following problems. If you do not get to these problems in class, you *must* do them before the quiz next week.

1. Define a relation on the integers by $a\equiv b (mod n)$ if $b-a=nk$ for some $k\in \mathbb{Z}$. Prove that this is an equivalence relation. Let ${\mathbb{Z}}/n$ be the set of equivalence classes under this relation. We proved in class that ${\mathbb{Z}}/n=\{ [0]_n,\ldots , [n-1]_n\}$.
2. Define a binary operation on ${\mathbb{Z}}/n$ by $[a]_n+[b]_n=[a+b]_n$.
   1. Is this binary operation well defined? That is, if $[a]_n =[a_1]_n$ and $[b]_n = [b_1]_n$, is $[a+b]_n = [a_1+b_1]_n$? (Make sure you understand why proving this proves that $+$ is well defined.)
   2. Prove that $([a]_n+[b]_n)+[c]_n=[a]_n+([b]_n+[c]_n)$.
   3. Prove that $([a]_n+[0]_n)=[a]_n=([0]_n+[a]_n)$.
   4. Prove that $[a]_n+[-a]_n=[0]_n=[-a]_n+[a]_n$.
   5. What can you conclude about the set ${\mathbb{Z}}/n$ under the binary operation $+$?