Q: prove or disprove that $\{(x^2, x^3): x \in \mathbb{Q}\}$ is a function

Question

I need to prove or disprove that the relation $R$ is a function.

$$R = \{(x^2, x^3): x \in \mathbb{Q}\}$$

If $R$ is a function - is it onto? Is it surjective?

I know that I need to check if $R$ is onto or to check if it's uniquely defined, but I'm not sure how to, exactly.

Answer 1

$R$ is not a function because for $x = 1$ we have $(1,1) \in R$ and for $x= -1$ we have $(1,-1)$ in $R$.

Therefore, $R$ would map $1$ to two different values.

Answer 2

It can't be a function because $(1,1) = (1^2,1^3)$ and $(1,-1) = ((-1)^2, (-1)^3)$ are both in $R$.

Answer 3

$$R=\{(x^2, x^3): x \in \mathbb{Q}\}$$ is not a function.

Note that both $(4,8)$ and $(4,-8)$ are in the set $R$.

In fact for every $q\in \mathbb{Q} $ , $(q^2, q^3)$ and $(q^2, -q^3)$ are in $R$ which makes it far from being a function.

Answer 4

$R$ is a function if for every $x^2$ there is a unique $(x^2,x^3)\in R$. However this doesnt hold, because for $x^2=1$ we have that $x=1$ or $x=-1$, and consequently $x^3=-1$ or $x^3=1$.