Assume that X is a countable infinite set Prove that there a

Assume that X is a countable infinite set. Prove that there are subsets A and B of X so that A B = , A B = X , and A and B are countable infinite.

Solution

Let X be the set of integers.clearly it is a countable infinite set.

let set A be set of even integers and set B be the set of odd integers.

clearly Aintersection B is empty set , however A union B is equal to original set \'X\' { as set of even integers + set of odd integers together will givwe set of Integers}

st A & set B both are countable Infinite set.

so we can depart Countable infinite set in two set such a way that there would be no common elements, but both the set would be countable infinite.