what is the Cardinality of subsets of Z of size 3

Question

I need to find out the cardinality of the subsets of Z of size 3,

|N| = א0 |R| = א, thats how we defined it in class.

My idea is to build a ZxZ matrix, we can see that the subsets of size 2 are all under the main diagonal. now we can build a path and for each step we name the current element we standing on the number of the step, we get a 1:1 and onto function from N --> ZxZ and we can conclude that the cardinality of the subsets of Z of size 2 is א0,

now we take all those ZxZ pairs we walked on and place them on another matrix Z x (ZxZ), and now we can make a new path, but the problem I encounter is that there are many many repetitions and elements such as (-1,-1,0) that cant be a subset.

can i ignore them and say : we number only valid and not seen already subsets? or is this approach not logical in the first place?

Thanks in advance for the help

Answer 1

You can do the standard trick of mapping $\{a_i\}$ into $\prod p_i^{a_i}$ where the $p_i$ are the primes to show that the cardinality of the finite subsets of Z is countable.

Answer 2

For each $n\in \mathbb N$, let $S_n$ be the set of subsets of $\mathbb Z$ with size $3$ where the largest absolute value of any element is $n$. For example, $S_1$ only contains $\{-1,0,1\}$, while $S_2$ has these nine subsets: $$ \{-2,-1,0\},\{-2,-1,1\},\{-2,0,1\},\\ \{-1,0,2\},\{-1,1,2\},\{0,1,2\},\\ \{-2,-1,2\},\{-2,0,2\},\{-2,1,2\} $$ Note that every size-three subset of $\mathbb Z$ appears in exactly one of the $S_n$ sets. Therefore, you can get an enumeration of all of the size-three subsets of $\mathbb Z$ by listing the elements of $S_1$, then listing the elements of $S_2$ after that, then listing the elements of $S_3$ after that, and so on ad infinitum. In particular, this proves there are countably many such subsets.

In general, writing a set $S$ as a countable union of finite sets is a good way to prove that $S$ is countable. This technique avoids the difficulty of explicitly describing a bijection with $\mathbb N$.