The cardinality of a union of two sets

Question

Assume that the cardinality of the union of two sets is continuum. How to prove that at least one of the sets has the cardinality of a continuum?

I suppose that it's possible to cope with it, using the operations with cardinals (for example, something like $\mathfrak{c}+\mathfrak{c}=\mathfrak{c}$), but i have no meaningful ideas.

Could you give me a hint, please?

Thank you in advance.

Answer 1

$Card ( A\cup B ) = \max \left(Card(A), Card(B)\right)$ for ordinals greater or equal to $\omega$. For finite ordinals (that is, if $A$ and $B$ are finite) this is obviously false as $Card ( A\cup B ) + Card ( A\cap B ) = Card(A) + Card(B)$ in this case.

Answer 2

If none of the two sets have cardinality bigger than $\mathbb{N}$, than both of the sets are countable. But the union of two countable sets is again countable. This leads to a contradiction.