Countability and Cantor's diagonal argument

The zigzag $z : \N \to \Z$ lists the integers as $z(0), z(1), z(2), \dots = 0, -1, 1, -2, 2, \dots$ What is $z(7)$?

Compute Cantor's pairing $\pi(2, 3) = \dfrac{(m + n)(m + n + 1)}{2} + n$ for $m = 2$, $n = 3$.

Which pair $(m, n)$ has $\pi(m, n) = 12$?

Which of these sets are countable? (Choose all that apply.)

A list begins $r_1 = 0.5123\ldots$, $r_2 = 0.3456\ldots$, $r_3 = 0.9995\ldots$. Using the lesson's rule (digit $n$ of $x$ is $4$ if digit $n$ of $r_n$ is $5$, and $5$ otherwise), how does $x$ begin?

After seeing the diagonal argument, Kofi says: “Fine, the list missed $x$. So put $x$ at the front of the list — now nothing is missing.” What is the best reply?

Let $A = \{1, 2, 3\}$ and define $f : A \to \mathcal{P}(A)$ by $f(1) = \{1, 2\}$, $f(2) = \{3\}$, $f(3) = \{1, 3\}$. List the elements of Cantor's diagonal set $D = \{a \in A : a \notin f(a)\}$.

How many elements does $\mathcal{P}(\mathcal{P}(\{1, 2\}))$ have?

Which Lean statement expresses Cantor's theorem for a type α?

Write pair(m, n) computing Cantor's $\pi(m, n) = (m+n)(m+n+1)/2 + n$, and its inverse unpair(k), which returns the pair (m, n) with pair(m, n) == k. (Hint: find the diagonal $d = m + n$ as the largest $d$ with $d(d+1)/2 \le k$.)