finding all the left cosets This is a homework problem:

Determine if the index $[G:H]$ is finite or infinite. List all the left $H-$ cosets in the following: 
$a.$ Let $G=\mathbb{Z}\times \mathbb{Z}, \ H=\{(x,x):x\in \mathbb{Z}\}.$
$b.$ Let $G=\mathbb{Z}\times \mathbb{Z}, \ H=\{(x,y):x,y\in \mathbb{Z}, 2|x,3|y\}.$
$c.$ Let $G=\mathbb{Q}, \ H=\mathbb{Z}.$  

For the first problem, I have tried this: 

Take $(m,n),(m',n')\in \mathbb{Z}\times \mathbb{Z}.$ Then, $$(m,n)+ H=(m',n')+H \ \Leftrightarrow (m-m',n-n')\in H \Leftrightarrow m-m'=n-n'.$$ From this, I think we can conclude that left H-cosets are the equivalence classes $[(m,n)]:=\{(m',n'):m-m'=n-n'.\}$ However, I cannot see the next step to list all the cosets. Any help will be appreciated.

When I can do the first one, I assume the rest can be done in similar way.  
 A: First, +1 for mentioning that this is homework. I'll stick to providing hints and let you fill in the gaps, since you seem to have a grasp of what to do.
For $\it{a.}$, if you rearrange what you've already worked out, then we see that the coset of $H$ containing $(m,n)$ depends only on the difference between $m$ and $n$. Can you use this to write down a list of elements, one in each possible coset?
For $\it{b.}$, let $(m,n)$ and $(m',n') \in G$. As you seem to understand already, we can rewrite the condition $(m,n) + H = (m',n') + H$ into conditions involving $m - m' \text{ mod }2$ and $n - n' \text{ mod }3$. Since "$x \text{ mod }2$" and "$x \text{ mod }3$" have only finitely many possible values, we expect only a finite number of cosets.
For $\it{c.}$, if $m + \mathbb{Z} = n + \mathbb{Z}$, then again we will get a condition on $m - n$. Note that for any $m \in \mathbb{Q}$, we can add an integer to $m$ to get a number in the interval $[0,1)$. How can we use this to write down a list of coset representatives?
