Often we are required to prove that a function f, is one-to-one, or an injection. A proof of this depends on how the function is given to us and what properties the function has. For functions that are given by some formula there is a standard way. We use the contrapositive of the definition of one-to-one, namely that if f(u) = f(v) then u = v. For example, suppose that is defined by where N denotes the positive integers. We start by assuming there are integers u and v for which f(u) = f(v). Thus, and so . Then, (u-v)(u+v) = 0. Since the product is zero then u-v = 0 or u+v = 0. The first condition tells us that u=v. The second condition says that v = -u. However, since u > 0 this would imply v < 0 which is impossible since v is positive. Hence, u =v as required and f is one-to-one. There are other methods used to prove a function is one-to-one. In calculus for example, if f is differentiable then it is sufficent to show that the derivative is positive everywhere or negative everywhere. In linear algebra, if f is a linear transformation it is sufficient to show that the kernal of f contains only the zero vector. If f is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice in the list.