# Suppose that S and T each have cardinality c. Show that $S\cup T$ also has cardinality c.

I tried to use the Cantor-Bernstein Theorem. First, we have $S\subset S\cup T$, so that $\left | S \right |\leqslant \left | S\cup T\right |$. This implies $\left | S\cup T \right |\geqslant c$. But I don't know how to prove the opposite direction. Could someone help me with it? Thank you very much!

Hint: $S=_c[0,1], T=_c[2,3]{}$.
• @LiangkaiHu I do mean that $|S\cup T|\leq |\mathbb R|=\mathfrak c$.But I don't get what you mean with $\mathfrak c$ being a finite number. – Git Gud Mar 19 '14 at 12:12