Show that this set is denumerable

Question

Let $A=\mathbb{Z}[x]_{deg\leq n}$ denote the set of all polynomials in $x$ with degree less than or equal to $n$. Show that A is denumerable.

I know that set of all polynomials with integer coefficients is countable but how can I prove this problem ?

How can you uniquely define a polynomial with degree less then $n$ using $n$ integers? — ItamarShmelo, Jan 10 '21 at 18:22
@ItamarShmelo yes it is. But I really dont get it. can you please give answer if possible I am very new in this topic. — Ali, Jan 10 '21 at 18:32

score 0 · Answer 1 · answered Jan 10 '21 at 18:23

0

$A$ is in bijection with $\mathbb Z^{n+1}$ which is countable as a finite product of countable sets.

answered Jan 10 '21 at 18:23

Can you please elaborate it a bit more – Ali Jan 10 '21 at 18:27
$\mathbb Z$ is countable, $\mathbb N \times \mathbb N$ is countable. And proceed then by induction. – mathcounterexamples.net Jan 10 '21 at 18:29
I really dont get it. can you please give answer if possible I am very new in this topic. Would really appreciate you help. – Ali Jan 10 '21 at 18:31
I suggest that you Google. Those are very common results. – mathcounterexamples.net Jan 10 '21 at 18:32
I have tried that already but couldnt find anything related to this problem. If you could find can you please give me the link ? – Ali Jan 10 '21 at 18:34

1 Answers1