Prove that the union of two denumerable sets is denumerableS

Prove that the union of two denumerable sets is denumerable.

Solution

Let us assume the two disjoint sets s and T, By using these we can make a proof of it

since S U T = S U (T - S)

With this assumption, this is no harder to do than proving that Z is equivalent to N.

Since S and T are denumerable, we may write

S = {s_1, s_2, ...} and T = {t_1, t_2, ...}.

Define a bisection f: N --> S U T

f(n) = s_{(n+1)/2} if n is odd

= t_{n/2} if n is even.

{s_1, t_1, s_2, t_2, ...}.