2 Prove that the set A x Z 7 divides x is denumerableSol

2. Prove that the set A = { x Z : 7 divides x } is denumerable.

Solution

Set A is set of all multiples of 7

Define the map from set of natural numbers, N to A

f(2m)=7(m-1),m=1,2,3,4.,,

f(2m-1)=-7m,m=1,2,3,4,....

This is a bijection from N to A

And hence A is denumerable ie countable