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I'm trying to find the cardinality of the set of all polynomials with coefficients in ℝ. What's wrong with the following proof:

Let $f$ be a function: $$f: \mathbb R[x] \to P(\mathbb R)$$ $$f(a_{0}+a_{1}x^{1}+...+a_{k}x^{k}) = \left \{ a_{0}, a_{1}, ..., a_{k} \right \}$$

For example: $$f(4.3x+2.5) = \left \{ 4.3,2.5 \right \}$$

f is obviously not injective, but is onto. Meaning that $$|\mathbb R[x]]| \geq \left | P(R) \right | $$

What am I missing?

bof
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Rizon
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  • Why is it onto? Can polynomials have an infinite number of coefficients? – Olivier May 16 '17 at 12:08
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    It doesn't look onto to me. What is a polynomial mapping to ${0,1,2,3,\ldots}$? – Joppy May 16 '17 at 12:08
  • @Joppy, $$f(3x^3+2x^2+1x^1+0) = {0,1,2,3}$$ – Rizon May 16 '17 at 12:15
  • @Olivier, what do you mean? Can a subset of R have an infinite number of elements? – Rizon May 16 '17 at 12:17
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    @Rizon - what about the rest of the natural numbers? I meant the (infinite) set of whole numbers $0,1,2,3,4,\ldots$. A power set of a set is all subsets, not just finite subsets. – Joppy May 16 '17 at 12:18
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    @Joppy, Oh! I admit I didn't know a subset can be infinite. Silly me :| – Rizon May 16 '17 at 12:26
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    You mean the cardinality of the set of all polynomials with real coefficients. The cardinality of a polynomial is nothing of interest. – martin.koeberl May 16 '17 at 12:53

2 Answers2

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Notice that the cardinality of polynomials of degree $0$ (Only free coefficients) is $|\mathbb{R}| = \mathfrak{c}$ (We just map such polynomials to their free coefficients).

The cardinality of polynomials of degree $1$ is $|\mathbb{R} \times \mathbb{R}| = \mathfrak{c}$.

The cardinality of polynomials of degree $2$ is $|\mathbb{R} \times \mathbb{R} \times \mathbb{R}| = \mathfrak{c}$

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The cardinality of polynomials of degree $n$ is $|\mathbb{R}^n| = \mathfrak{c}$

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Your set is just the countable union of all these sets from above, and therefore its cardinality is also $\mathfrak{c}$. (See here: Cardinality of union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$)

Henno Brandsma
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AsafHaas
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Ok so look at the interval $[-n,n]$.Total number of subsets of it is $c$...now.coefficients come from one of these subsets for some $n \in N$. So it is less than $c* \aleph $...which is still $c$.The reverse inclusion is trivial.So cardinality of total polynomial is $c$.

CoffeeCCD
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    -1 for € and alepnot. Please use MathJax (in which case I'll remove my -1): https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – martin.koeberl May 16 '17 at 12:52
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    Is it okay now? Typing from comp is ok from mobile its difficult – CoffeeCCD May 16 '17 at 13:03