(a) (b) (c) (d) (e) As a general result, the set of all functions from a set $A$ of cardinality $\alpha$ ($\alpha$ is either a positive integer if $A$ is finite or $\omega$ if $A$ is countably infinite) to a set $B$ is in a bijective correspondence with $B^{\alpha}$ (there is a bijection between them, see Exercise 7 of §6). Hence, we get countably infinite sets in (a), (b) and (c) (using Corollary 7.4, Theorem 7.5 and Theorem 7.6), and uncountable sets in (d) and (e) (using Theorem 7.7).

(f) (g) To continue the general result stated above, the set $F$ of all functions from $\mathbb{Z}_{+}$ to a set $B$ that are eventually a given fixed element $b\in B$ is in a bijective correspondence with the countable union over $n\in\mathbb{Z}_{+}$ of the sets $F_{n}$ all functions from $S_{n}$ to $B$ s.t. $f(n)\neq b$ (where $F_{1}=\{\emptyset\}$ ). Indeed, having a sequence $(b_{1},\ldots,b_{m},b,b,\ldots)$ where $b_{m}\neq b$ , we map it to the function $f:S_{m+1}\rightarrow B$ such that $f(i)=b_{i}$ , and, hence, $f(m)\neq b$ . It is clear that the mapping is injective and surjective. Now, $F_{1}$ is a singleton, $F_{2}:\{1\}\rightarrow B-\{b\}$ is in a bijective correspondence with $B-\{b\}$ , and for $n\ge3$ , each $F_{n}$ is clearly in a bijective correspondence with $B^{n-2}\times(B-\{b\})$ . So, if $B$ is countable, so is each $F_{n}$ for $n\ge2$ , and their countable union together with $F_{1}$ is countably infinite. And this is what we get in both (f) and (g).

(h) We further continue our result, the set of all functions from $\mathbb{Z}_{+}$ to a set $B$ that are eventually constant is simply the union over $b\in B$ of the sets of all functions from $\mathbb{Z}_{+}$ to $B$ that are eventually $b$ . Hence, if $B$ is countable, we have the countable union of countably infinite sets (according to (f) and (g)), which is countably infinite.

(i) Let $f:I\rightarrow A$ be such that $f(\{a,b\})=g\in A$ where $g(0)=\min\{a,b\}$ and $g(1)=\max\{a,b\}$ . $f$ is clearly injective, and $A$ , according to (a), is countably infinite. So is $I$ (it is infinite as it contains, for example, all two-element sets of the form $\{n,n+1\}$ for $n\in\mathbb{Z}_{+}$ ).

(j) Similar to (i), if $J_{n}$ is the set of all $n$ -element subsets of $\mathbb{Z}_{+}$ , then $J_{0}$ contains one subset only, and for $n\ge1$ , there is an injective correspondence $f_{n}:J_{n}\rightarrow B_{n}$ such that $f_{n}(\{a_{1},\ldots a_{n}\})=g\in B_{n}$ where $g(1)<g(2)<\ldots<g(n)$ and $g(\{1,\ldots,n\})=\{a_{1},\ldots,a_{n}\}$ . Since, according to (b), for all $n\in\mathbb{Z}_{+}$ , $B_{n}$ is countably infinite, so is $J$ (including $J_{0}$ ).