Cardinal numbers

We know that, the relation in sets defined by A~ B is an equivalence relation. Hence by fundamental theorem on equivalence relation, all sets are partitioned into disjoint classes of equivalent sets.

Thus for any set A, equivalence class of A, [A] = { B | B ~ A }

Result: - (1) [A] = [B] or [A] ∩ [B] = ∅ , that is for any two sets, either they have same equivalence classes or totally disjoint equivalence classes.

(2) = universal set.

(3) If A ∈ [B] then A ~ B (4) If A ∉ [B] then A ≁ B

Definition :- Let A be any set and let α denote the family of sets which are equivalent to A. Then α is called a Cardinal number (or, simply cardinal) of the set A, i.e., the cardinal number of A is the equivalence class of A. Cardinal number of A is denoted by or card (A) or #(A).

∴ The Cardinal number α of the set A is the collection of all sets equivalent to A, i.e., α = #(A)

The cardinal number of each of the sets ϕ, {1}, {1, 2}, {1, 2, 3} is denoted by 0, 1, 2, 3 respectively, and is called a finite cardinal.

The cardinal numbers of N ( the set of natural numbers), and R (the set of real numbers) are denoted by

#(N) = 0 (read aleph-null), #(R) = c

For finite sets the cardinal number is easy to describe. We shall say that any set equipotent to the set {1, 2, 3, . . . , n} has cardinal number n. Thus for finite sets, the cardinal number of a set is just how many points it contains.

Among the infinite sets, the denumerable ones will be said to have cardinal number 0. Sets which are equipotent to the set of all real numbers is said to have cardinal number c.

In mathematics, we do not generally use the definition of cardinal numbers. It is enough to know, if two cardinal numbers are equal or less than, etc. For example, #(A) = # (B) iff A ~ B ( [A] = [B] )

# (I) = 0 , # (Q) = 0 (∵ and )

Order relation in cardinals

Let α = card (A) , β = card (B)

We say that α ≤ β if A is equipotent with a subset of B

i.e., ∃ f : A ⟶ B1 , which is one-one & onto where B1 ⊆ B

We say that α < β if α ≤ β & α ≠β

Theorem:- The order relation is antisymmetric that is α ≤ β & β ≤ α ⟹ α = β

Sum of Two cardinals : -

Suppose α, β are the given cardinals.

Choose two disjoint sets A & B such that A ∈ α & B ∈ β , then we define α+ β = card ( A ∪ B )

e.g., 3 = # ( { a, b, c } ) and 4 = # ( { 1, 3, 5, 7 } ). Then 3 + 4 = # ( { a, b, c, 1, 3, 5, 7 } ) = 7

Multiplication of cardinals :- Let α & β be the cardinal numbers and let A & B be disjoint sets such that

α = # (A) , β = # (B) , then α β = # ( A × B )

e.g., 3 = # ( { a ,b , c} ) and 4 = # ( {1, 3, 5, 7 } ). Then (3) (4) = # ( { a, b, c } × { 1, 3, 5, 7 } ) = 12

Thus the operations of addition & multiplication of finite cardinal numbers corresponds to the ordinary operations of addition & multiplication of natural numbers.

Above definitions of sum & multiplication are well defined, i.e., the definitions of α+ β & α β do not depend upon the particular sets A & B. In other words, if

A ~ A’ , B ~ B’ , A ∩ B = ϕ , A’ ∩ B’ = ϕ then # ( A ∪ B ) = # ( A’ ∪ B’ ) & # ( A × B ) = # ( A’ × B’ )

i.e., A ∪ B ~ A’ ∪ B’ & A × B ~ A’ × B’

Theorem:- The operations of addition & multiplication of cardinal numbers are associative and

commutative ; and addition distributes over multiplication , i.e., for any cardinal numbers α, β & γ (i) (α + β) + γ = α + ( β + γ ) (addition is associative) (ii) (α β) γ = α (β γ) (multiplication is associative) (iii) α +β = β + α (addition is commutative) (iv) α...