which of the following are sublattices in Z^2? what is the index??

1.{(x,y) in Z^2 s.t. x+y=1}

2.{(x,y) in Z^2 s.t. x+y=0}

3.{(x,y) in Z^2 s.t. 2|x}

4.{(x,y) in Z^2 s.t. 2x-2y=0(mod 3)}

5.{(x,y) in Z^2 s.t. 2x-2y=0(mod 6)}

A subgroup of Rn is called a (full) lattice in Rn if L = Zv1 +Zv2 + + Zvn for some linearly independent vectors v1, v2 ,..., vn of Rn. For

example, Zn is lattice in Rn.

A sublattice of Zn is a subgroup of finite index.

i dont know how to consider the index . I tried to think about the left coset, but it seems every set can have infinite choices of x and y , many thanks for any hins or suggestion