Real analysis - Countable and uncountable set

Question

I'm having a problem understanding this:

The union of a countable set and an uncountable set is uncountable.

Help me please!

Answer 1

Hint: Let $C$ be countable and $U$ be uncountable. Can you show that $U\subseteq C \cup U$?

Answer 2

If the union of a countable set $A$ and an uncountable set $B$ were countable, then $B$, being a subset of the countable set $A \cup B$, would be countable, a contradiction.

Answer 3

Since a union contains all elements of both sets, it must have cardinality at least as great as the larger set (uncountable in this situation).

Answer 4

Say $A$ is a countable set and $B$ is uncountable.

If $A\cup B$ was countable than there was an injection $f:A\cup B \to \mathbb{N}$.

But than if you define $g:B \to \mathbb{N}$, $g(x) = f(x)$, than $g$ is also an injection and $B$ is countable. Contradiction.

Intuitively, It's impossible to gather 2 sets together and get a smaller cardinality than one of the sets.