Set-builder notation, quantifiers and infinite sets

List the elements of $\{n \in \N : n^2 < 30\}$. (Remember that $\N = \{0, 1, 2, \dots\}$.)

Which sentence is the denial (negation) of “Every train from Mombasa arrived on time today”?

Is the statement “Every element of the empty set is greater than $100$” true or false?

Which of these statements are true? Choose all that apply.

Let $S$ be the set of students in a class, and let $M(t, s)$ mean “$t$ has met $s$”. Which statement says “There is a student whom every student in the class has met”?

Which sentence is the denial of “Every country has a city with more than a million people”?

How many elements does the set $\{x \in \Z : -3 \le 2x + 1 \le 7\}$ have?

Lena wants to prove that the even numbers are infinite. She is given a non-empty finite list of even natural numbers whose largest entry is $M$. Which number is certainly an even number missing from the list, whatever the list is?

Which Lean statement says that there is no largest natural number?

Pierre de Fermat believed (around 1640) that every number $2^{2^k} + 1$ is prime. Write first_counterexample(prop, limit), which returns the first k in range(limit) for which prop(k) is False, or None if there is none. Then use it with the is_prime function provided to test Fermat's claim for $k = 0, 1, \dots, 5$.