I want to show that the Cardinalities of R[X] and R are equal. For starters, I tried mapping each polynomial to Rn by mapping it to its coordinates vector, but its not correct (I am limiting my polynomials here)

Humberto Campbell

Humberto Campbell

Answered question

2022-11-04

Cardinalities of R[X] and R
I want to show that the Cardinalities of R[X] and R are equal.
For starters, I tried mapping each polynomial to R n by mapping it to its coordinates vector, but its not correct (I am limiting my polynomials here)
I can use cardinalities arithmetic and Cantor - Bernstein theorem.
Would love to get some insight.

Answer & Explanation

martinmommy26nv8

martinmommy26nv8

Beginner2022-11-05Added 16 answers

Step 1
The set of polynomials of degree n is clearly in bijection with R n , which again is equinumerous to R, or with the open interval ] n , n + 1 [.
Step 2
Then R[X] can be injected into n ] n , n + 1 [ R
Jonas Huff

Jonas Huff

Beginner2022-11-06Added 3 answers

Step 1
R n [ X ] the set of polynomials of degree n has the same cardinal than R n + 1 , which has for cardinal the cardinal of R. This is the consequence that for an infinite set X, the cardinality of X × X is the one of X.
Step 2
Then R [ X ] = n R n [ X ] and therefore the cardinality of R[X] is equal to the one of R.

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