Math 360, Fall 2015, Assignment 2

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

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

1. Reflexive (relation).
2. Symmetric (relation).
3. Transitive (relation).
4. Equivalence relation.
5. Equivalence class.
6. Partition (of a set $A$).
7. Binary operation (on a set $A$).
8. Commutative (binary operation).
9. Associative (binary operation).

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

1. Theorem relating equivalence classes to partitions (this is Theorem 0.22 in the book).

Solve the following problems:

1. Section 0, problems 29, 30, and 31.
2. Section 1, problems 1, 2, 3, 4, 7, 8, and 9.

--------------------End of assignment--------------------