EDB — 1YW

3.10 Cardinality[1YW]

[22B]↺↻

In the following, let \(E_ 0=∅\), and let \(E_ n=\{ 1, \ldots n\} \) otherwise if \(n ≥ 1 \).

Note that the null map \(f: ∅ → ∅ \) is a bigection; and \(|A|=0 ⇔ A=∅\). The following exercise is a fundamental result.

We recall Theorem 1.12.2 of the notes [ 3 ] , for convenience.

Finite sets

E28

[02T]↺↻

E28

[02W]↺↻

E28

[02Y]↺↻

E28

[22K]↺↻

Comparison

E28

[030]↺↻

E28

[031]↺↻

E28

[032]↺↻

E28

[034]↺↻

E28

[036]↺↻

E28

[038]↺↻

E28

[03C]↺↻

E28

[03F]↺↻

Countable cardinality

E29

[03H]↺↻

E29

[03M]↺↻

E29

[03P]↺↻

E29

[03R]↺↻

Cardinality of the continuum

[ [03W]↺↻]

E31

[03X]↺↻

E31

[03Y]↺↻

E31

[040]↺↻

E31

[043]↺↻

E31

[045]↺↻

In general

Let’s add some more general exercises.

E31

[048]↺↻

E31

[04B]↺↻

E31

[04D]↺↻

E31

[04G]↺↻

E31

[04J]↺↻

E31

[22M]↺↻

E31

[04M]↺↻

E31

[04P]↺↻

E31

[04R]↺↻

E31

[04V]↺↻

E31

[04X]↺↻

E31

[04Z]↺↻

E32

[051]↺↻

E32

[053]↺↻

E32

[055]↺↻

E32

[057]↺↻

Other interesting exercises are [0T9]↺↻, [0MZ]↺↻.

Power

Recall that \(A^ B\) is the set of all functions \(f:B→ A\). We will write \(|2^ A|\) to indicate the cardinality of the set of parts of \(A\).

E32

[05J]↺↻

E32

[05M]↺↻

In general in case \(|B|{\gt}|A|\) the study of the cardinality of \(|B^ A|\) is very complex (even in seemingly simple cases like \(A=ℕ\)).

[ [05Q]↺↻]