I am not looking for direct answers unless asked for as I am trying to understand the material and improve my discrete mathematics knowledge, thank you.
Proposition: Sets $A, B$ and $C$ are disjoint denumerable sets. Prove that $A \cup B \cup C$ is also denumerable.
What I know: I know that a denumerable set has the same cardinality (size) of the set of Natural Numbers ($\mathbb N$). I also know that $A$, $B$, and $C$ are disjoint sets meaning they have no common elements so hence: $A \cap B \cap C = \emptyset$
If we were to list out the set of elements contained in $A \cup B \cup C$ we would have a set (In this example I use $Z$) $Z = \{a_1, b_1, c_1, a_2, b_2, c_2, \ldots \}$.
Is this enough to prove the proposition? We can clearly see this set is infinite as well meaning it must have the same cardinality of the set $\mathbb N$. If not how else can this be re-written in order to show a more concise proof and what other ways could I have re-written the set $Z$ used in the example above.
And that's the key.
But you need rigor.
$A$ is denumerable. Meaning there is a bijection $\phi_a:\mathbb N\to A$ so that $\phi_a(i) = a_i$ so we can list $A$ as $A = \{a_1, a_2, a_3....\}$.
Likewise as $B$ and $C$ are denumerable there are similar bijections $\phi_b: \mathbb N\to B$ and $\phi_c: \mathbb N\to C$ so that $B = \{\phi_b(1), \phi_b(2), ....\} = \{b_1, b_2,...\}$ and $C = \{\phi_c(1), \phi_c(2), ....\} = \{c_1, c_2,...\}$
So FORMALLY define a bijection $\psi: \mathbb N \to A\cup B\cup C=Z$ where $\psi (i) = x_i = .... $ some specific element of $Z=A\cup B \cup C$.
Now bear in mind $\psi(1) = a_1=x_1$ and $\psi(2) = b_1 = x_2$ and $\psi(3) = c_1 = x_3$ and $\psi(4) = a_2 = x_4$.
Can you formally define $\psi$? And once you define $\psi$ can you prove it is a bijections? If so, that is all you need to do.