injection

n 1: the forceful insertion of a substance under pressure

2: any solution that is injected (as into the skin) [syn: {injectant}]

3: the act of putting a liquid into the body by means of a

syringe; "the nurse gave him a flu shot" [syn: {shot}]

injection

1. 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. 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)