Sum of two countably infinite sets What is the sum of two countably infinite sets? Another countably infinite set? I am asked to find this in a question.
 A: Hint: Assuming "sum" means "union", let $A = \{a_1, a_2, \dots\}$ and $B = \{b_1 , b_2, \dots\}$ be two countable sets.  Consider the map
$$
f(n) = \left\{
\begin{array}{ccc}
b_{n/2} & & n \text{ is even}
\\
a_{\frac{n+1}{2}} & & n \text{ is odd}
\end{array}
\right.$$
Show that $f(n)$ is a bijection from $\mathbb{N} \to A \cup B$.
A: Spoiler:
Can you count their elements?
1,3,5,7...
2,4,6,8...  
A: For the operation suggested in the question, the proper name is 'union', not sum. The sum of two sets (the Minkowski sum) can be defined as 
$$ A+B=\{a+b : a \in A, b \in B\}.$$
In this case, it can be proved that if $A,B$ are countable, then so is $A+B$. First, note that $A \times B \simeq \Bbb{N}\times \Bbb{N}\simeq \Bbb{N} $ ( $ \simeq$ means that they have the same cardinal number).
Then the function $f: A\times B \to A+B $ defined by $f(a,b)=a+b$ is surjective by definition. This means that $\operatorname{card}(A \times B) \geq \operatorname{card}(A+B)$, therefore $A+B$ is countably infinite also.
