1. Go to the following survey
2. Fill it out with your thoughts on the class. I REALLY appreciate honest feedback, and will make changes next term in response to your thoughts, so please take time to honestly reflect on your experience. The long-response questions near the bottom are the most helpful ...

1. Consider a graph with $5$ points, where every point is connected by a line to every other point. (A graph like that is sometimes called a complete graph with 5 points, or $K_5$)
   1. How many lines are there within this graph?
   2. How many triangles (groups of 3 points, all of ...

1. Prove, using induction, that $ 7^n - 1 $ is always divisible by $6$, as long as $n$ is an integer.

Another way of writing that that expression is divisible by $6$ is to say that $ 7^n - 1 = 6x$, where $x$ is some integer. ($x$ is the whole number you would ...

1. I have a bunch of identical pairs of gloves jumbled together in a box. Suppose I take 5 gloves out at random. The pigeonhole principle guarantees that I will get at least one Left handed glove and at least one Right handed glove.

Is the preceding statement true or false ...

1. Prove, using induction, that $ 1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}$

1. State De Morgan's Laws
2. Prove one of De Morgan's Laws (you choose) using a truth table
3. If you roll one die (with 6 sides) four times, writing down the number each time,
   1. How many different strings of numbers could you possibly get?
   2. What if the dice are not ...

1. In a tournament with $\mathbf{n}$ teams, if every team plays each of the other teams exactly once, and every team wins at least once, then at least two of the teams will end up with the same win-loss record. Explain why.

Induction

1. State DeMorgan's Laws
2. How many different ways are there to arrange 7 books on a shelf?
3. Explain what you have to do, physically, to solve the book-stacking problem for 6 books, in which you are trying to move a stack of 6 books from one table to another,following ...

1. Is the formula $(x+5)^2 = x^2 + 25$ always true? If it is, find a way to prove that it is true. If it is not, provide a counterexample (a value of $x$ which shows that the formula is not always true).
2. Write down all the subsets of the ...

There is a crazy mailman who does not like to put more than $\mathbf{x}$ letters in a mailbox. If there are $\mathbf{b}$ mailboxes and $\mathbf{L}$ letters, is it possible for him to put all the letters in the boxes without exceeding $\mathbf{x}$ letters in any box ...

If there are 8 people at a party,

1. How many possible ways are there for all of them to line up (to get drinks, for example)?
2. How many possible ways are there to pick four of those people out of the drink line, keeping them in order?
3. How many ways ...

$\mathbf{S}$: It is snowing. $\mathbf{Q}$: We are having a quiz.

1. Construct truth tables for the following statements and compound statements: $\mathbf{S}, \mathbf{Q}, \mathbf{S}\vee\mathbf{Q}, \mathbf{S}\wedge\mathbf{Q}, \mathbf{Q} \to \mathbf{S} $
2. Which case(s) is(are) true today?

Combinatorics

More techniques of combinatorics

1. What conclusions can you draw, given the premises below:

   1. $ \mathbf{P \to Q}$, $\mathbf{P}$
   2. $ \mathbf{P \to Q}$, $\mathbf{\lnot Q}$
   3. $ \mathbf{A \to B}$, $\mathbf{B \to C}$

2. If $x$ is evenly divisible by 2 and $y$ is evenly divisible by 2, then $x$ is evenly divisible by ...