Which sets defines a function Let $B$ and $C$ be two arbitrary sets. A set $A\subset B\times C$  defines a function $f: G(f) \to Y$, if:  $(x,y_1),(x,y_2) \in A \Rightarrow y_1 = y_2$.

$G(f) = \{x \in X \mid \exists y \in Y$ such that $ (x,y) \in A\}$ and $f(x) = y \to (x,y) \in A$.

$a) A_1 := \{(x,y) \in \mathbb{R}\times\mathbb{R}: x = y^2\}\text{ or } A_2 := \{(x,y)\in \mathbb{R}\times\mathbb{R}: x = y^3\}$

Which set in $a)$ defines a function and is it surjective or injective?

Every time I think I understood something in math, it hits me so hard for being so naive... Intuitively I would say set $A_1$ is a function. I do understand the symbols and terms but I do not see the relations..
No. $A1$ is not a function. Note that both $(1,1)$ and $(1,-1)$ are elements of $A1$.
No. $A1$ is not a function. Note that both $(1,1)$ and $(1,-1)$ are elements of $A1$.