injection

1. mathematics A function, f : A -@# B, is injective or

one-one, or is an injection, if and only if

for all a,b in A, f(a) = f(b) =@# a = b.

I.e. no two different inputs give the same output (contrast

many-to-one). This is sometimes called an embedding. Only

injective functions have left inverses f' where f'(f(x)) = x,

since if f were not an injection, there would be elements of B

for which the value of f' was not unique. If an injective

function is also a surjection then is it a bijection.

2. reduction An injection function is one which takes

objects of type T and returns objects of type C(T) where C is

some type constructor. An example is

f x = (x, 0).

The opposite of an injection function is a projection

function which extracts a component of a constructed object,

e.g.

fst (x,y) = x.

We say that f injects its argument into the data type and fst

projects it out.

(1995-03-14)