Course Outlines

Denumerable sets; Countable sets; Uncountable sets; Cardinality of infinite sets; Cardinal arithmetic;


Cantor's theorem, Schoeder Bernstein theorem; and Continuum hypothesis.


Introduction 

One to one correspondence: A one to one correspondence between the set A and the set B is defined as a pairing of elements of A with the elements of B such that  

i)    Each element of A is paired with exactly one element of B

ii)   Each element of B is paired with exactly one element of A.

      Equivalent sets

 The set A is said to be equivalent to set B, denoted by A~B, if there is a one – one 

correspondence between the sets A and B.

 Less power and greater power

 Let A and B be two sets. If the set A is equivalent to a certain subset of set B, but the set B is not equivalent to any subset of the set A, then the set A is of less power than the set B, denoted by |A|<|B|. The set B is greater power than the set A.

Denumerable or countably infinite set

 An infinite set is said to be denumerable or countably infinite if it is equivalent to the set of natural numbers.

Countable set

 A set is said to be countable if it is either finite or denumerable.

Infinite set

 A set is said to be infinite set if it is equivalent to the proper subject of itself.

Uncountable set

 A set is said to be uncountable if it is neither finite nor denumerable set.

 Finite and transfinite cardinal numbers

The cardinal number of a finite set is called the finite cardinal number. The cardinal number of an empty set is 0, the cardinal number of singleton set is 1 and 2 is the cardinal number of set equivalent to {a, b}.

   The cardinal number of an infinite set is called transfinite cardinal number. The cardinal number corresponding to the denumerable set is denoted by À0 and cardinal number corresponding to non-denumerable set is denoted by c.

Some theorems regarding to finite and transfinite cardinal numbers.

i) n + À0 = À0, n Î À.          ii) À0 + À0 = À0                         iii) À0 + À0 + ... + À0 = À0

iv) À0 + c = c                               v) c + c = c                                vi) c + c0 + ... + c = c

vii) À0 × À0 = À0                       viii) c × c = c