Fundamentals of Pure Mathematics

Prefatory remarks 

Fundamentals of Pure Mathematics 

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Alan J. Cain (Rm. 331) Fundamentals of Pure Mathematics 2006 1 / 10

P(N) 

More cardinality and countability 

Theorem 24.1 

|(0, 1)| = |P(N)|. (So P(N) is uncountable.) 

Proof. 

Define f : (0, 1) → P(N) by 

0.a 1 a 2 a 3 ... ↦→ {i : a 

} {{ } 

i = 1}, 

binary expansion 

where a j ∈ {0, 1}. 

f is an injection, so |(0, 1)| ≤ |P(N)|. 


P(N) 


Proof (cont.) 

Define g : P(N) → (0, 1) as follows, 

g(X) = 0.a 1 0a 2 0a 3 ... 

} {{ } 

binary expansion 

(i ∈ X =⇒ a i = 1, i /∈ X =⇒ a i = 0). 

g is an injection. 

So, by the Schröder–Bernstein theorem, |(0, 1)| = |P(N)|. 

Why not g(X) = 0.a 1 a 2 a 3 ... Because then 

g({1}) = 0.1000 ... = 0.0111 ... = g(N − {1}). 

(cf. 0.1000 ... = 0.0999 ... for decimal expansions.) 



More countable sets 

Proposition 24.2 

Let P be the set of all subsets of N of size 2. Then P is countable. 


Clearly |P| ≤ |N × N| = |N|. Since N has the smallest cardinality of any 

infinite set, |P| = |N|. 



More countable sets 


Let P n be the set of all subsets of N of size n. For any n ∈ N, P n is 

countable. 


Clearly |P n | ≤ |P n−1 × N|. By induction, P n−1 is countable. So 

|P n | ≤ |N × N| = |N|. So |P| = |N|. 



Finitary power set 

Recall that PX, the power set of X, is defined by 

PX = {Y : Y ⊆ X}. 

The finitary power set of X, denoted P F X, is the set whose members 

are finite subsets of X 

P F X = {Y : Y ⊆ X, Y is finite} 

Notice that 

P F N = {∅} ∪ ⋃ n∈N 

P n . 



Finitary power sets 


P F (N) is countable. 



Countable unions of countable sets 

Theorem 24.5 

If S i is countable for each i ∈ N, then 

S = ⋃ i∈N 

S i 

is countable. 


Let f i : S i → N be a bijection. Define f : S → N × N by 

x ↦→ (i, f i (x)) 

where x ∈ S i but not in S 1 ∪ · · · ∪ S i−1 . 

This map is an injection, so |S| ≤ |N × N| = |N|. So |S| = |N|. 



Finitary power sets 

Proof of Proposition 24.7. 

P F (N) is the union of the various countable sets P n and the 1-element 

set {∅}, so is itself countable. 


Fin 


This is the end of the course, but we have a few matters to clear up: 

Surprise Egg Competition winners 

Feedback survey 

Next week — revision lectures on Monday and Tuesday (Jan 2007 

paper) 

Next week — tutorial & examples class

Fundamentals of Pure Mathematics

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