1)is the ring of real polynomials R[X] countable and why ?
To determine whether the ring of real polynomials, denoted as R[X], is countable, we need to understand the concept of countability and apply it to the given set.
Countability refers to the cardinality of a set, which is the measure of its size or how many elements it contains.
A set is countable if its elements can be put into a one-to-one correspondence with the set of natural numbers (N) or any subset of N. In simpler terms, a set is countable if its elements can be listed using a systematic method.
To determine whether R[X] is countable, we can use a proof by showing that it can be put into a one-to-one correspondence with a countable set. In this case, we can show that R[X] is countable by associating each polynomial with a unique tuple of coefficients.
Here's a step-by-step explanation of how we can prove that R[X] is countable:
1. Start by defining a bijection between R[X] and R^n, where n represents the degree of the polynomial (the highest power of X).
2. Since R^n is countable (as each coordinate of the tuple can be associated with a natural number), we can say that R[X] is countable if it can be put into a one-to-one correspondence with R^n.
3. For each polynomial in R[X], we can associate it with an n-tuple of its coefficients, where the i-th coefficient represents the coefficient of X^i in the polynomial. This establishes a one-to-one correspondence between R[X] and R^n.
4. Therefore, since R^n is countable, we can conclude that R[X] is also countable.
In summary, the ring of real polynomials R[X] is countable because it can be put into a one-to-one correspondence with a countable set, specifically R^n, where n represents the degree of the polynomial.