Proof Question: Set $\theta = \{((x^2,y),(y,2x,x-y)): x,y \in \mathbb{Z}\}$ a function? What is domain, range, codomain?

Question

Set $\theta = \{((x^2,y),(y,2x,x-y)): x,y \in \mathbb{Z}\}$ a function? What is domain, range, codomain?

Domain: set of $(x^2,y)$ for $x,y \in \mathbb{Z}$

Range: set of $(y,2x,x-y)$ for $x,y \in\mathbb{Z}$ and the codomain is the same as the range.

This is a function because each set is "unique" from the previous set.

If I did this all correctly?

Answer 1

Note that $$((1^2,0),(0,2 \cdot 1,1-0)) = ((1,0),(0,2,1))$$ and $$(((-1)^2,0),(0,2 \cdot (-1),-1-0)) = ((1,0),(0,-2,-1))$$ both belongs to $\theta$.