Answer
a. $P(x) :=$ x belongs to a set that is a proper subset of $\mathbb{Q}$ and $\mathbb{R}$.
b. $P(x) :=$ x belongs to a set that is denumerable.
Work Step by Step
a. The key is to think at the level of sets, not the elements. Since $\mathbb{Z}$ is a subset of $\mathbb{Q}$ and $\mathbb{R}$, any property that integers satisfy is also satisfied by the rational and real numbers. What sets $\mathbb{Z}$ apart as a set is that it is the proper subset of both $\mathbb{Q}$ and $\mathbb{R}$. $\mathbb{Q}$ is a proper subset of $\mathbb{R}$ but it is not a proper subset of itself by definition of proper subset. $\mathbb{R}$ is neither a proper subset of $\mathbb{Q}$ nor of itself.
b. The one property that differentiates between $\mathbb{Z}$ and $\mathbb{Q}$ on one hand and $\mathbb{R}$ on the other hand is denumerability i.e., the property of having the same cardinality as $\mathbb{N}$. $\mathbb{Z}$ and $\mathbb{Q}$ are denumerable whereas $\mathbb{R}$ is non-denumerable.
