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Cardinality and Countability

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Cardinality and Countability

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Now · 9.1 Comparing Sizes

Cardinality and Countability

9.1 Comparing Sizes

Sets

A

and

B

have the same cardinality (

|A| = |B|

) if there exists a bijection

f: A \to B

|A| \leq |B|

if there exists an injection

f: A \to B

9.2 Countable Sets

A set is countable if it is finite or has the same cardinality as

\mathbb{N}

. Examples of countable sets:

$\mathbb{Z}$ — map $n \to 2n$ (positive) and $n \to -(2n+1)$ (negative)
$\mathbb{Q}$ — diagonal enumeration of fractions
$\mathbb{N} \times \mathbb{N}$ — use Cantor pairing function $\pi(m,n) = \frac{(m+n)(m+n+1)}{2} + n$

Countable Union Theorem

A countable union of countable sets is countable.

9.3 Uncountable Sets

\mathbb{R}

is uncountable — there is no bijection with

\mathbb{N}

Cantor's Diagonal Argument

Assume

f: \mathbb{N} \to \mathbb{R}

is a bijection. List all real numbers in

(0,1)

f(1) = 0.a_{11}a_{12}a_{13}\dots

f(2) = 0.a_{21}a_{22}a_{23}\dots

f(3) = 0.a_{31}a_{32}a_{33}\dots

Construct

b = 0.b_1b_2b_3\dots

where

b_i \neq a_{ii}

(e.g.,

b_i = 5

a_{ii} \neq 5

b_i = 6

a_{ii} = 5

). Then

b

differs from every

f(i)

at the

i

-th decimal place, so

b

is not in the list. Contradiction.

\square

9.4 Schröder-Bernstein Theorem

|A| \leq |B|

and

|B| \leq |A|

, then

|A| = |B|

✅ Practice Questions

Q1: Show

[0,1]

and

(0,1)

have the same cardinality.

Solution

Construct a bijection: Map $0 \to \frac{1}{2}$ , $\frac{1}{2} \to \frac{1}{3}$ , $\frac{1}{3} \to \frac{1}{4}$ , etc., and map all other points to themselves. This is a bijection between $[0,1]$ and $(0,1)$ . Q2: Is the set of all infinite binary sequences countable? Solution

No. By diagonal argument: assume countable, list them, flip the $i$ -th bit of the $i$ -th sequence. The resulting sequence is not in the list. So uncountable. Join Discord PreviousAdvanced Logic & InductionNextUncountable Sets