Question 12
Let $\mathbb{Z}$ be the set of integers and $\mathbb{Z} \times \mathbb{Z}$ the set of ordered pairs of integers. The sum of these pairs is defined by $$
(a, b)+\left(a', b'\right)=\left(a+a', b+b'\right)
$$ being $(-a,-b)$ the opposite of $(a, b)$. Study if there is a subset $E$ of $\mathbb{Z} \times \mathbb{Z}$ that meets the following conditions:
a) The sum of two pairs of $E$ is also from $E$.
b) The $(0,0)$ pair belongs to $E$.
c) If $(a, b)$ is not $(0,0)$, then either $(a, b)$ belongs to $E$, or $(-a,-b)$ belongs to $E$, but not both.
(a, b)+\left(a', b'\right)=\left(a+a', b+b'\right)
$$ being $(-a,-b)$ the opposite of $(a, b)$. Study if there is a subset $E$ of $\mathbb{Z} \times \mathbb{Z}$ that meets the following conditions:
a) The sum of two pairs of $E$ is also from $E$.
b) The $(0,0)$ pair belongs to $E$.
c) If $(a, b)$ is not $(0,0)$, then either $(a, b)$ belongs to $E$, or $(-a,-b)$ belongs to $E$, but not both.