Is the relation a function?

Question

Is the relation on $\Bbb {R}$ a function from $\Bbb {R}$ to $\Bbb {R}$?

$$\{(a^2,a)\mid a \in \Bbb {R}\}$$

How do I determine whether or not the relation is a funtion? Would I treat $(a^2,a)$ as $(x,y)$?

Answer 1

If relation $R$ is a function then $(a,b)\in R\wedge (a,c)\in R$ implies that $b=c$.

This is not the case for $R:=\{(a^2,a)|a\in\mathbb R\}$.

For instance we have $(4,2)\in R$ and $(4,-2)\in R$ while $2\neq-2$.

Answer 2

No, because if you consider $-1$ and $1$, you get $$(1,1)=(1^2,1)\in R$$ as well as $$(1,-1)=((-1)^2,-1)\in R$$

Where R is the relation you defined.

Thus, your relation gives two values for one argument at a certain point which is invalid for a function.

This would indeed not be the case if you switched the positions: The relation $\{(a,a^2)|a\in \mathbb{R}\}$ is a valid function.