ページ1:

§ 4.4 Bijective Functions
Def
A function f A→ B
is
called
bijective iff it is
surjective
and
injective.
Ex
Of A B
f(1): := d
2
B
A = {1, 2, 3, 4}
f(2) == 8
3
B = {α, A, 8, 8}
B,
B,
f(3):= B
4
-8
f(4) = 8
•) f: 2 → 2
f(z) =
= 2 +1.
Surjective
:
Let
WEZ
f(w) =
www.
2.
= W.
(Vy 6B: 3 XEA: (x))
(VwEZ:FzEZ: (2, w) Ef.).
(Z=W-1)
injective: Assume f(z) = f(22). ↔ 8₁+1 = 82+| ↔ 2₁ = 82.
=
•) f: R + R.
f(x) = 3x+1.
↑
bijective
Thm
Recull
Let f A B
→>
that f* = {(xx) (x, y) ε f}
Then
f
is bijective
There
exists a 9: BA St.
gof=IA
^
fog= IB.
Rmk
In this
case
g=f".
In
particular,
the function 9 is unique.

ページ2:

Ex
Ex
9:
Ex
• g: B → A
g (α) := |
gof(1) = g(f(1)) = g(x)= |
9(B) == 3
gof(2) = 9 (f(2)) = 9(8)=2
9(8) : = 2
918)=4
fog (α) = f(g(a)) = f(i)=9.
9 (w) w-l.
(F(2)-2+1)
go f(z) = g(2+ 1) = (2+1)−1 = 2.
fog (2) = f (2-1) = (2-1) +1 =8
g: R R
g(y)
=:
y-l
3
y= 3x+1. < 4-1 = 3x <
y-1
= X
3
proof of the
Assume f is bijective
Set g:= f"
g
is actually
9
function.
wts
Vyeb
: 3 x A
(y, x) Eg
proof
(x, y) = f (=) f(x) =y.
Since ₤ surjective and injective
Vy EB: 3! XEA: (x,y) Ef. <) VYEB: 3: KEA: (y, x) Eg_
g is
a
well-defined function.