2011-05-19, 08:53 PM
A function is, by definition, one-to-one because we already have a term for a relation which is not necessarily one-to-one.
What you're asking is akin to something like, why does a square have to have equal sides? Rectangles don't have equal sides and they're perfectly good geometric figures.
It's useful to have a name for a one-to-one relation. Many ideas can be expressed in terms of functions. At any given location, the temperature at that location is a function of time. You can't have two temperatures at the same time.
Edit: "One-to-one" actually refers to a function that is invertible, that is, only one input for every output. Otherwise, it would be redundant terminology.
http://en.wikipedia.org/wiki/Injective_function
What you're asking is akin to something like, why does a square have to have equal sides? Rectangles don't have equal sides and they're perfectly good geometric figures.
It's useful to have a name for a one-to-one relation. Many ideas can be expressed in terms of functions. At any given location, the temperature at that location is a function of time. You can't have two temperatures at the same time.
Edit: "One-to-one" actually refers to a function that is invertible, that is, only one input for every output. Otherwise, it would be redundant terminology.
http://en.wikipedia.org/wiki/Injective_function