Chapter 1.9, Problem 7E

Let $A$ and $B$ be two nonempty sets. If there is an injection of $B$ into $A$, but no injection of $A$ into $B$, we say that $A$ has greater cardinality than $B$.

(a) Conclude from Theorem $9.1$ that every uncountable set has greater cardinality than ${\mathbb{Z}}_{+}$.

(b) Show that if $A$ has greater cardinality than $B$, and $B$ has greater cardinality than $C$, then $A$ has greater cardinality than $C$.

(c) Find a sequence $A_1,A_2,\mathrm{...}$ of infinite sets, such that for each $n\in {\mathbb{Z}}_{+}$, the set $A_{n+1}$ has greater cardinality than $A_n$.

(d) Find a set that for every $n$ has cardinality greater than $A_n^{}$.