## Math 110-02, Spring 2014 Information on the Final Exam

The last test for this course will be given during the scheduled final exam period: Sunday, May 11, at 1:30 PM. The exam will be in our regular classroom. The final exam counts for 15% of the overall grade for the course.

The exam will be five or six pages long, and it will be only a little longer than the three in-class tests from earlier in the semester. There will be one essay question at the end covering general ideas from the course, and you will know that essay question in advance. Aside from that essay question, the exam is not cumulative. It will cover material from the last part of the course, since the third test. This includes: dimension and the fourth dimension; graphs, including Euler circuits, the Euler characteristic, and regular polyhedra; and voting, including various voting methods and Arrow's Impossibility Theorem. The reading from the textbook includes sections 4.7, 6.1, 6.2, and 10.4; however, we did a few things that are not in the book.

Here is the general essay question for the end of the test: "Over the course of the semester, we have discussed many mathematical ideas, but one that came up over and over is the idea of infinity. Write an essay discussing infinity and what you have learned about it, including some of the specific ways that infinity has come up in the course. What in the end do you think about the mathematical idea of infinity?"

Here are some other terms and ideas that might be on the test:

```the fourth dimension

understanding the fourth dimension by analogy
what a 2D object looks like passing through a 1D world (that is, a line)
what a 3D object looks like passing through a 2D world (that is, a plane)
what a 4D object might look like passing through a 3D world (that is, space)
imagining living on the surface of a torus or in the 3D analog of a torus
how a torus can be modeled as a rectangle with edges identified
how a 3D torus can be modeled as a brick or fishtank with sides identified.

hypercube (also known as tesseract)

graph (in the sense of vertices plus edges)
vertex (plural is "vertices")
edge
understanding diagrams of graphs
how a graph can be used to model connections by bridges between land masses
Euler circuit
degree of a vertex (number of edges that have the vertex as an endpoint)
connected graph
a graph has an Euler circuit if and only if
it is connected and all vertices have even degree
finding an Euler circuit in a graph
Euler path
a graph has an Euler path if and only if it has an Euler circuit
OR is connected and has exactly two vertices that have odd degree
finding an Euler path in a graph

regular polygon
regular polyhedron  (plural is "polyhedra")
Platonic solids:  tetrahedron, cube, octahedron, dodecahedron, icosahedron
the formula  V - E + F  for graphs and polyhedra:  How to count V, E, F
planar graph (drawn in the plane so that edges don't cross)
for any planar graph, V - E + F = 2
for any polyhedron (without "holes"), V - E + F = 2
the only regular polyhedra are the five Platonic solids

social choice (how a group can make a choice among alternatives)
voting
ranking (each voter lists alternatives in order of preference)
voting methods:
plurality voting
plurality voting with runoff
IRV (Instant Runoff Voting)
Borda count
approval voting
Condorcet winner: wins every one-on-one matchup with other alternatives

desirable properties of voting methods
1.  No dictator:
The winner is not simply the choice of some particular voter.
2.  Unanimity:
If one alternative is the first choice of every voter,
then that alternative wins.
3.  Ignore the irrelevant:
The result doesn't change if a losing alternative drops out.
4.  Better is better:
If some voters raise their ranking of the winning alternative, that
will not cause that alternative to lose.

Arrow's Impossibility Theorem:
There is no voting method that satisfies the four above properties.
```