Show that the set of all even integers is infinitely countab

Show that the set of all even integers is infinitely countable.

Solution

Let E be the set of even integers and f(x) = 2x be a function from N to E.       

             1    2    3    4    5     6  …..   

2    4    6    8    10  12  ……

Then f is a bijection from N to E since f is both onetoone and onto.  

To show that it is injective, f(n) = f(m).Then 2n  = 2m, and so n = m.

To see that it is surjective, suppose that t is some even positive integer. Then t = 2k for some positive integer k and f(k) = t.