What is the cardinality of the following sets? (Choose from finite, countably infinite, or uncountably infinite.)
The set of polynomials of the form $ax+b$ with $a \in\Bbb N$ and
- $b \in\{0,1\}$
- $b \in\Bbb N$
- $b \in\Bbb Q$
- $b \in\Bbb R$
- $b \in\Bbb C$
2026-04-02 23:57:12.1775174232
cardinality of polynomial
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First, is the domain of $x$ all real numbers or is it relative to each of the sets listed out in (1) - (5)?
After thinking about this, ask yourself the following questions:
(1) Given that $a \in \mathbb{N}$, and that $b \in \{0,1\}$, does adding a set of finite elements to that of an infinite set change the total cardinality of the union of both sets?
(2) Does the union of two sets with cardinality = $\mathbb{N}$ equal to $\mathbb{N}$ or something else?
(3) Are the rationals equinumerous to the naturals?
(4) Are the Reals equinumerous to the naturals? (Hint: read about Cantor Diagonalization)
(5) How are the Complex Numbers defined? Does their definition, coupled with your answer to question (4), dictate the cardinality?