In: Advanced Math
Define
, for any
,
where
is the set
of all natural numbers.
Recall that a set S is called a finite set if there exists
,
be such there is a bijection between S and
and a set
is called an infinite set if it is not finite.
Also recall if denotes
the cardinality of a set then,
, for all n. and for any two sets M and N if there exists an
injective map
, the we will
have
.
A set is called countable infinite if there is a bijection
between the set and .
Given that , where
is
countable infinite. To show B is an infinite set.
Proof by contradiction method. Let's assume by contrary that B
is a finite set. Then by definition we can find ,
such that B is bijection with
Since
, there is
an injective map from A to B, Thus we get an injective map from A
to
Let f be the
bijectiion between A and
, then
, is also a bijective, and A to
, is an injective,
thus by composing we get an injective map from
,
thus this means
, which is a contradiction. Hence B is an infinite set.