Math 360, Fall 2016, Assignment 1

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. Cartesian product (of two sets).
2. Relation (from $A$ to $B$).
3. Function (from $A$ to $B$).
4. Domain (of a function).
5. Codomain (of a function).
6. Image (of a function).
7. Injection (a.k.a. one-to-one function).
8. Surjection (a.k.a. onto function).
9. Bijection.
10. Equinumerous (a.k.a. equipotent or having the same cardinality).

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

1. Russell's paradox (this is not exactly a theorem, but it is an important fact).
2. Cantor's theorem.

Solve the following problems:

1. Section 0, problems 1, 5, 7, 11, and 12.
2. Prove that the function $f:\mathbb{Z}\rightarrow2\mathbb{Z}$ defined by the formula $f(n) = 2n$ is a bijection. (Hint: to prove injectivity, assume that $f(a) = f(b)$ and show that $a=b$. To prove surjectivity, let $b$ be any element of $2\mathbb{Z}$, and find some $a\in\mathbb{Z}$ with $f(a) = b$.)

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