p(b) = some a X such that f (a) = b. It does not matter which a we choose, but there will be such an a by denition of f [X ]. We are placing the members of f [X ] in the pigeonholes X . By the pigeonhole principle, some pigeonhole has at least two occupan
(m1 , m2 ) = (n1 , n2 ), but f (m1 , m2 ) = f (n1 , n2 ). There are many possibilities, such as (1, 0) and (0, 1). In fact, since + is commutative, (m, n), (n, m) is a counter-example for any m, n. EXAMPLE 4.10 The function f on natural numbers dened by f
EXAMPLE 4.6 Let A = cfw_1, 2, 3 and B = cfw_a, b. The function f = cfw_(1, a), (2, b), (3, a) is onto, but not one-to-one, as is immediate from its diagram:
A
1 2 3 a b c
B
It is not possible to dene a one-to-one function from A to B , as there are too ma
The partial function f is regarded as undened on those elements which do not have an image under f . It is sometimes convenient to refer to this undened value explicitly as (pronounced bottom). A partial function from A to B is the same as a function from