Math 360, Fall 2019, 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

Read:

1. Section 2.

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

1. Equinumerous (sets).
2. Finite (set).
3. Countable (set).
4. Binary operation (on a set $S$).
5. Binary structure.
6. Commutative (binary structure).
7. Associative (binary structure).
8. Identity element (in a binary structure).
9. Inverse (of an element of a binary structure with identity).
10. $\equiv_n$ (the relation of congruence modulo $n$).

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

1. Cantor's theorem (concerning whether the sets $\mathbb{Z}$ and $\mathbb{R}$ are equinumerous).
2. Theorem concerning the properties of the congruence relation (i.e. "For any positive integer $n$, the relation $\equiv_n$ is...")

Solve the following problems:

1. Section 0, problems 16 and 17.
2. Section 2, problems 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, and 20.
3. Let $n$ be any positive integer. Let $n\mathbb{Z}$ denote the set of integer multiples of $n$, and 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.
4. Prove that $n\mathbb{Z}$ is countably infinite, for any positive integer $n$.

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