# Cardinality of set of functions with coefficients from a set with cardinality omega

Let $A_1$, and $A_2$ be subsets of $\mathbb{R}$ of cardinality $\aleph_0$, $\aleph_1$ respectively. Let $P_1$ be the set of polynomials of the form $a_n(x^n) + a_{n−1}(x^{n−1}) +··· a_1x + a_0$ where $n < \omega$ is any natural number and the coeﬃcients $a_i$ all come from $A_1$, and $P_2$ is deﬁned similarly with the $a_i(i \leq n)$ all coming from $A_2.$ What are the cardinalities of $P_1$ and $P_2$?

$$|P_1|=\left|\bigcup_{n\in\omega}A_1^n\right|=\aleph_0\cdot\aleph_0=\aleph_0$$
$$|P_2|=\left|\bigcup_{n\in\omega}A_2^n\right|=\aleph_0\cdot\aleph_1=\aleph_1$$