What is a one-to-one relation in mathematics?

Answer 1

A one-to-one relation in math means that for every value in the range, there's at most one value that maps to it. If you think of relations as people sitting on a bus, this means that no one is sharing a seat. More often, mathematicians talk about one-to-one functions (formally called injections) - these are just one-to-one relations that happen to be functions :-). If we write the function as y=f(x), the condition for it being one-to-one is that if f(a)=f(b), then a=b. When looking at the graph of a relation, we can determine if it's one-to-one by the horizontal line test: if any horizontal line drawn on the graph intersects the relation at most once, it is one-to-one. On the other hand, if a horizontal line intersects the relation twice, the relation is not one-to-one. For example, y=x^2 is not a one-to-one relation: (-1)^2 and 1^2 both get mapped to 1. We can see this in the graph of y=x^2 because a horizontal line above the x-axis will intersect the graph twice. y=x^3, on the other hand, is a one-to-one relation. Because the cube of a negative number stays negative, no two numbers get sent to the same number by cubing them.