Notation and definitions
Let f : X → Y. The pre-image of y ∈ Y is the set of x ∈ X such
that f (x) = y. This is Definition 9.1.2 in the book. I use the standard
notation f −1 (y) = {x ∈ X | f (x) = y} for this pre-image.
Let Nn + {1, 2, . . . , n} and let Z2 + {0, 1} (in general, Zn + {0, 1, . . . , n−
1}.
The characteristic function of a subset A ⊆ X is the function χA :
X → Z2 is defined by
{
0 if x ∈
/ A,
χA (x) =
1 if x ∈ A.
This defined in #15 in Problems II.
IX denotes the identity function from X to X, i.e., IX (x) = x for
x ∈ X. This is defined in Example 8.1.4 in the book.
Proof that |P(Nn )| = |Fun(Nn , Z2 )|
Let P(Nn ) be its power set of all subsets of Nn , that is, A ∈ P(Nn ) if
and only if A ⊆ Nn . Let Fun(Nn , Z2 ) be the set of functions from Nn to Z2 ,
that is, f ∈ Fun(Nn , Z2 ) if and only if f : Nn → Z2 .
Theorem. The function b : P(Nn ) → Fun(Nn , Z2 ) defined by b(A) = χA
is a bijection. Hence |P(Nn )| = |Fun(Nn , Z2 )| .
Proof. Define the function c : Fun(Nn , Z2 ) → P(Nn ) by
c(f ) = f −1 (1) = {k ∈ Nn : f (k) = 1}
for f : Nn → Z2 .
We shall show that c is the inverse of b.
Step 1 (c ◦ b = IP(Nn ) )
c (b (A)) = c(χA ) = χ−1
A (1) = A.
Justification of the last equality: since χA (k) = 1 if and only if k ∈ A, we
have
χ−1
A (1) = {k ∈ Nn : χA (k) = 1}
= {k ∈ Nn : k ∈ A}
= A.
1
Step 2 (b ◦ c = IFun(Nn ,Z2 ) )
(
)
b (c (f )) = b f −1 (1) = χf −1 (1) = f.
Justification of the last equality: Let k ∈ Nn . Then f (k) = 0 or f (k) = 1.
(0) Suppose f (k) = 0. Then f (k) ̸= 1 implies k ∈
/ f −1 (1). So
χf −1 (1) (k) = 0 = f (k) .
(1) Suppose f (k) = 1. Then k ∈ f −1 (1). So
χf −1 (1) (k) = 1 = f (k) .
We have proved for any k ∈ Nn that χf −1 (1) (k) = f (k) . Hence χf −1 (1) =
f.
2