# Properties of unions of sets

1) Prove that the union of a finite set and a countable set is countable. I am thinking that a finite set by definition has the cardinality of {1,2,...,n} and a countable set has the cardinality of natural numbers, so both have the cardinality of natural numbers? I don't see how to prove this rigorously, as it seems to be very basic intuition?

2) Prove that the union of two countable sets is countable. Again, can (or why not) this be proven by the same logic as 1) ?

• HINT: Any subset of a countable set is also countable. Not all countable sets are finite. – CAGT Jan 20 '14 at 21:37
• Have you put any effort into searching for the question on the site before posting it? – Asaf Karagila Jan 20 '14 at 21:37

To demonstrate the first, think about listing the $n$ things in your finite set, and then listing the countably many things in your infinite set: If you can write your infinite set as $A = \{a_1, a_2, a_3, ...\}$ and your finite set $B = \{b_1, ..., b_n\}$, can you write down a bijection between
$$\{b_1, b_2, ..., b_n, a_1, a_2, a_3, ...\}$$ and the set $\mathbb{N}$?
Likewise, given two countable sets, try interlacing elements, e.g. $\{a_1, b_1, a_2, b_2, ...\}$ and see if you can biject it with $\mathbb{N}$.