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?