My textbook says that there is Cartesian product $A \times B$ which produces yet another set of all possible pairs $(a \in A, b \in B)$. So far so good. Than something really unexpected happens:
Subsets of $A \times B$ are called 'relations'. We will define a mapping or a function $f \subset A \times B$ from set $A$ to set $B$ to be the special type of relation in which for each element $a \in A$ there is a UNIQUE element $b \in B$ such that $(a, b) \in f$.
Seems like $f(x) = x^2$ is not a function anymore nor a mapping? Since any positive output "is mapped" with two rather than strictly one input, i.e. $2^2 = 4, -2^2 = 4$.
Wouldn't it be more correct to define mapping as injective function also known as "one-to-one" function? So "maps" are more strict kind of functions?
Mappings and functions are the same.
By "unique $b$", the definition is saying that $a$ appears first in exactly one tuple $(a,b)$ (i.e. each $a$ doesn't have two different $b$'s $b_1$ and $b_2$ for which $(a, b_1)$ and $(a, b_2)$ are both in the relation).
It does not necessarily mean that for any $a_1$ and $a_2$, if $(a_1, b_1)$ and $(a_2, b_2)$ are in the relation, then either $a_1=a_2$ or $b_1 \neq b_2$. (In other words, it is okay for each $b$ to correspond to more than one $a$).
In the example of $f(x)=x^2$, some outputs, or $b$'s, have more than one $a$ mapped to them, but each input, or $a$, has exactly one $b$ to which it is mapped.