I am learning basic set theory and was doing some exercises where we are to determine whether a given relation is a function, an injection, a surjection and if it is a bijection.
The question in this case was: Determine whether the relation from $\mathbb{R}^2$ to $\mathbb{R}$ defined by $(x,y)Rz$ if and only if $z=x^2+y^2$ is $(1)$ a function, $(2)$ an injection, $(3)$ a surjection and $(4)$ a bijection.
$(1)$ is clearly true. $(2)$ It is not an injection as $(x,y)$ and $(y,x)$ both map to the same point in $\mathbb{R}$. $(3)$ Neither is it a surjection as $x^2+y^2\ge0, \forall x,y\in\mathbb{R}$. $(4)$ This leads me to believe that it is therefore not a bijection. However the answers says it is. This is strange as the book defines a bijection to be a function which is both an injection and a surjection. Clearly the given relation does not satisfy either condition so cannot be a bijection.
I think the answers might have some typos though since it is missing the answer to the next exercise (A relation from $\mathbb{Z}\times\mathbb{Z}$ to $\mathbb{Z}\times\mathbb{Z}$ where $(a,b)R(x,y)$ if an only if $y=a$ and $x=b$; which I found to be a bijection) and so might have mixed up the answer to this exercise with the previous. However, I am new to set theory and so was wondering whether I just made a silly mistake.