Math 360, Fall 2016, Assignment 1
From cartan.math.umb.edu
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:
- Cartesian product (of two sets).
- Relation (from A to B).
- Function (from A to B).
- Domain (of a function).
- Codomain (of a function).
- Image (of a function).
- Injection (a.k.a. one-to-one function).
- Surjection (a.k.a. onto function).
- Bijection.
- Equinumerous (a.k.a. equipotent or having the same cardinality).
Carefully state the following theorems (you do not need to prove them):
- Russell's paradox (this is not exactly a theorem, but it is an important fact).
- Cantor's theorem.
Solve the following problems:
- Section 0, problems 1, 5, 7, 11, and 12.
- Prove that the function f:Z→2Z 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 2Z, and find some a∈Z with f(a)=b.)