Relations, equivalence relations and partitions

On the set $\{0, 1, 2, \dots, 11\}$, let $a \sim b$ mean that $a$ and $b$ have the same remainder on division by $4$. List the elements of the class $[6]$.

Which properties does the relation $a \le b$ (“$a$ is less than or equal to $b$”) on $\Z$ have? Choose all that apply.

On $\Z$, let $a \approx b$ mean that $a$ and $b$ differ by at most $2$. This relation is not an equivalence relation. Which of these is a genuine counterexample?

On the whole numbers $0, 1, 2, 3, \dots$, let $a \sim b$ mean “$a$ and $b$ have the same last digit”. How many equivalence classes are there?

The partition $\{\{a, b, c\}, \{d\}, \{e, f\}\}$ of $\{a, b, c, d, e, f\}$ defines the relation “lies in the same block as”. How many ordered pairs does this relation contain?

Here is a “proof” that every symmetric, transitive relation is reflexive.

Let $a$ be any element. (1) Choose $b$ with $a \sim b$. (2) By symmetry, $b \sim a$. (3) By transitivity, from $a \sim b$ and $b \sim a$ we get $a \sim a$.

What is wrong?

Which of these is a partition of $\{1, 2, 3, 4, 5, 6\}$?

How many different equivalence relations are there on the set $\{1, 2, 3, 4\}$? (Use Theorem 3: count the partitions instead.)

In the Lean section, cls R a x means “$x$ belongs to the class of $a$”. Which statement says that any two classes are equal or disjoint?

Write is_equivalence(elements, related), where elements is a list and related(a, b) returns True or False. It should return True exactly when the relation is reflexive, symmetric and transitive on elements. Python's all(...) is handy: all(related(a, a) for a in elements) tests reflexivity. What should your function return for an empty list, and why?