Question

Give an example to prove that intersection of denumerable sets need not be denumerable

Answer 1

consider the set

$A=\{ 2,4,6,8,...........\}$

$=$ set of all even natural numbers

$B=\{ 1,3,5,7,.............\}$

$=$ Set of all odd natural numbers

Now define maps

$f:\N \rightarrow A \ and \ g:\N\rightarrow B$

By $f(n)=2n \ and \ g(n)=2n-1$

Clearly , $f \ and \ g$ are one-one and onto .

Therefore , $A \ and \ B$ are denumerable.

But $A\cap B=\empty$ , which is not an infinite set .

Hence $A\cap B=\empty$ is not a denumerable set.