Math 360, Fall 2017, Assignment 2

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

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

1. Equivalence relation.
2. Equivalence class.
3. Partition.
4. Function.
5. Injective (function).
6. Surjective (function).
7. Bijective (function).
8. Equinumerous (sets).

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

1. Statement relating equivalence relations to partitions.
2. Cantor's theorem (concerning whether the sets $\mathbb{Z}$ and $\mathbb{R}$ are equinumerous).

Solve the following problems:

1. Section 0, problems 12, 15, 16, 17, 23, 24, and 25.
2. Section 0, problems 29, 30, 31, and 32 (in these problems, you have already determined whether the relation in question is an equivalence relation in the previous assignment; now for those that are, go ahead and describe the partition into equivalence classes).
3. Define a map $f:\mathbb{Z}\rightarrow2\mathbb{Z}$ by the formula $f(x)=2x$. Show formally that this map is both injective and surjective.

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