Mathematical Analysis I, 2004a

§§4–7. Relations. Mappings 11
A mapping f is said to be “from A to B” iffD f = A and D
f ′
write
f : A → B (“f maps A into B”).
⊆ B; wethen
If, in particular, D f = A and D
f ′
write
f : A −→ B
onto
= B, wecallf a map of A onto B, andwe
(“f maps A onto B”).
If f is both onto and one to one, we write
f : A ←→
onto
B
(f : A ←→ B means that f is one to one).
All pairs belonging to a mapping f have the form (x, f(x)) where f(x) is
the function value at x, i.e., the unique f-relative of x, x ∈ D f . Therefore, in
order to define some function f, it suffices to specify its domain D f and the
function value f(x) for each x ∈ D f . We shall often use such definitions. It is
customary to say that f is defined on A (or “f is a function on A”) iff A = D f .
Examples.
(a) The relation
R = {(x, y) | x is the wife of y}
is a one-to-one map of the set of all wives onto the set of all husbands.
R −1 is here a one-to-one map of the set of of all husbands (= D ′ R )onto
the set of all wives (= D R ).
(b) The relation
f = {(x, y) | y is the father of x}
is a map of the set of all people onto the set of their fathers. It is not one
to one since several persons may have the same father (f-relative), and
so x ≠ x ′ does not imply f(x) ≠ f(x ′ ).
(c) Let
g =
( )
1 2 3 4
.
2 2 3 8
Then g is a map of D g = {1, 2, 3, 4} onto D ′ g
= {2, 3, 8}, with
g(1) = 2, g(2) = 2, g(3) = 3, g(4) = 8.
(As noted above, these formulas may serve to define g.) It is not one to
one since g(1) = g(2), so g −1 is not a map.