Math 360, Fall 2021, Assignment 2

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

Carefully define the following terms, then give one example and one non-example of each:

1. Binary relation (from $A$ to $B$).
2. Reflexive (binary relation).
3. Symmetric (binary relation).
4. Anti-symmetric (binary relation).
5. Transitive (binary relation).
6. Equivalence relation.
7. Equivalence class (of an element $a\in A$, with respect to an equivalence relation $\sim$ on $A$; also known as $\left[a\right]_\sim$).
8. Partition (of a set $A$).
9. $S/\sim$ (the partition arising from the equivalence relation $\sim$ on the set $S$).
10. $\equiv_n$ (the relation of congruence modulo a non-negative integer $n$).
11. Function (from $A$ to $B$).
12. Domain (of a function).
13. Codomain (of a function).
14. Image (of a function).

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating equivalence relations to partitions.
2. Theorem concerning the key properties of $\equiv_n$ (i.e. "$\equiv_n$ is an...").

Solve the following problems:

1. Section 0, problems 12, 23, 25, 29, 30, 31, 32, and 33.
2. Calculate the cardinality (i.e. the number of elements) of $\mathbb{Z}/\equiv_n$. Illustrate your calculation with a concrete example, listing the elements of $\mathbb{Z}/\equiv_n$ explicitly.
3. A binary relation which is reflexive, anti-symmetric, and transitive is called a partial ordering. Give at least one example of a partial ordering. (Hint: you may wish to ignore the word "partial," which functions here mainly as a distraction.)
4. Now looks for an example of a partial ordering which shows why we should call them partial orderings in general.

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