This chapter puts in formal terms every child's instinctive activity of sorting a collection of things into different kinds. We will say that things count as being of the same kind when they are related by an equivalence relation, or they are equivalent. When we have gathered together all things of the same kind we call the set an equivalence class.

Concurrent reading: Green, chapter 2.

1. 1 Two lines in a plane may be parallel, perpendicular or neither. If l ∥ m, does it follow that m ∥ l? If l ⊥ m, does it follow that m ⊥ l?

2. These results make both parallelism and perpendicularity symmetric relations.

3. Is orthogonality of circles a symmetric relation?

4. Is the divisibility of natural numbers a symmetric relation?

5. 2 If l ∥ m and m ∥ n, does it follow that l ∥ n? (We allow a line to be parallel to itself.) If l ⊥ m and m ⊥ n, does it follow that l ⊥ n?

6. These results make parallelism, but not perpendicularity, a transitive relation.

7. Is orthogonality of circles a transitive relation?

8. Is divisibility of natural numbers a transitive relation?

9. 3 If a set P is partitioned into disjoint subsets A1, A2, A3, …, where Ai ∩ Aj = ∅ unless i = j, we call the Ai, parts of P and the set of all Ai a partition of P.

10. […]