Math 360, Fall 2018, Assignment 3

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. Partition.
2. Function.
3. Domain (of a function).
4. Codomain (of a function).
5. Image (of a function).
6. Injective (function).
7. Surjective (function).
8. Bijective (function).
9. Equinumerous (sets).
10. Inverse relation (of a relation from $A$ to $B$).
11. Inverse relation (of a function from $A$ to $B$).
12. Invertible function.
13. Forward image (under a function, of a subset of its domain).
14. Pre-image (under a function, of a subset of its codomain).
15. Fiber (of a function, over a point in its codomain).

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).
3. Theorem relating invertibility to bijectivity.
4. Theorem relating bijectivity to fibers.

Solve the following problems:

1. Section 0, problems 12, 15, 17, 23, 24, and 25.
2. Let $n$ be any positive integer. Define a map $f:\mathbb{Z}\rightarrow n\mathbb{Z}$ by the formula $f(x)=nx$. Show formally that this map is both injective and surjective.
3. Prove that $n\mathbb{Z}$ is countably infinite, for any positive integer $n$.
4. Let $f:A\rightarrow B$ be any function, and consider the following two statements: (1) For any $C\subseteq A$, one has $f^{-1}[f[C]]=C$, and (2) For any $D\subseteq B$, one has $f[f^{-1}[D]]=D$. Give examples to show that both of these statements are false in general. (In class I spoke carelessly concerning this matter.)
5. (Challenge) Try to correct the two statements from the previous problem, and prove your corrected versions.

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