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**HallsofIvy** Plato's method works, of course, but I had a different idea. Since A is countable, there exists a function N->A that is one-to-one and countable so each member of A can be "labeled" $\displaystyle a_n$. Similarly, members of B and C can be "labeled" $\displaystyle b_n$ and $\displaystyle c_n$. Now define a function from N to the union of A, B, and C by $\displaystyle f(3n)= a_n$, $\displaystyle f(3n+1)= b_n$, and $\displaystyle f(3n+2)= c_n$. It is easy to show that function is bijective.