Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from 2003 to 2012. Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. You may be able to find more up-to-date versions of some of these notes at http://www.cs.yale.edu/homes/aspnes/#classes.

1. Surjectivity

Let f:A→B. Prove that f is surjective if and only if, for all sets C and functions g,h:B→C, g∘f = h∘f implies g = h.

2. A recursively-defined function

Let f:ℕ→ℤ be defined by the rule:

f(0) = a
f(1) = b
For n > 1, f(n) = 2f(n-1) - f(n-2)

where a,b∈ℤ.

Show that there exist constants c and d (which may depend on a and b but not on n), such that f(n) = cn+d for all n∈ℕ.

3. Sums of products

Prove that the following identity holds for all n∈ℕ:

$1 + \sum_{k=1}^{n} \left(k \cdot \prod_{i=1}^{k} i\right) = \prod_{i=1}^{n+1} i.$