I am very new to proofs so please excuse any trivial errors.
In lecture we were told that:
A set $\mathbb S$ is called finite if there exists a one-to-one mapping (bijective mapping) between $\mathbb S$ and the set $\mathbb N_n=\{0,1,2,3,...,n-1\}$ for some natural number $n$. A set is called countable infinite if there exists a bijective mapping between $\mathbb S$ and the set $\mathbb N=\{0,1,2,...\} $ of all natural numbers
I want to answer the following questions:
1. Show that the set $\mathbb S=\{a,b,c\}$ is countable
Does countable mean countable infinite? An attempt from another post on this site: For all triples $(a,b,c)$ there exists an according sum $S=a+b+c \implies ...$ But wouldn't the sum be a surjection between $\mathbb S$ and $\mathbb N$ and not a bijection?
2. Show that the set of integer numbers $\mathbb Z=\{0,\pm1,\pm 2,...\}$ is countable infinite
Before I continue with this proof I have a question about mapping. The definition says that "if there exists a bijective mapping...". In this context, can a function $f(x)$ be considered a mapping? If yes, then can't I just define myself some map $$f:\mathbb Z\to \mathbb N$$
and let $\space f$ be: $$x \mapsto \lvert x \rvert$$
which would then automatically imply that $\mathbb Z$ is countable infinite? (or wouldn't it)?
3. Show that $\mathbb S=\mathbb N \times \mathbb N=\{x |x=(p,q),p,q \in \mathbb N\}$ is countable infinite.
I am not sure about this one but if my "intuition" from 2) turns out to be correct I can maybe define some sort of norm and show that it is a bijection between $\mathbb S$ and $\mathbb N$