Let A ⊆ ℤ with 0 ∈ A. For r, s ∈ ℤ, define
rA = {ra : a ∈ A}, rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group ℤ?
1
−2A ⊆ A, A + A = A
2
A = −A, A + 2A = A
3
A = −A, A + A = A
4
2A ⊆ A, A + A = A