a State the definition of countable b State the definition o

(a) State the definition of countable.

(b) State the definition of denumerable.

(c) Prove, using your definition above: If A and B are denumerable and disjoint, then

A B is denumerable.

Solution

a) A given set A is countable if there exists a bijection between A and a subset of natural numbers

b) A set A is denumerable if there exists some bijection between A and the set of natural numbers

c) A is denumerable so there is some bijection,f, from A to N

And some bijection,g,from B to N

Define a map:

h from A union B to N

For a in A

h(f(a))=2f(a)

h(f(b))=2g(b)-1

This is clearly 1-1 and onto

Hence A union B is denumerable.