# Mathematics project

Solve all of the problems below, and provide full justification and explanations for every answer.
Your project must be typed, and should not be longer than two or three pages. You may not
work on these problems with other students or use any outside resource, but you are
welcome to ask questions and discuss them with me either during class or in office hours. The point
of this assignment is to practice your technical writing skills – you should convince me that your
solution to each problem is correct!
Problem 1. Exampletown is a city made up of two kinds of people – nurts and blargs. Nurts
always tell the truth, and blargs always lie. Every person is either a nurt or a blarg.
(a) There are two people, A and B. A says “At least one of us is a blarg.” Determine if each of A
and B is a nurt or a blarg.
(b) C says “Either I am a nurt or D is a blarg.” Determine if each of C and D is a nurt or a blarg.
(c) E says “I am a blarg, but F isn’t.” Determine if each of E and F is a nurt or a blarg.
(d) There are three people, G, H, and J. G says “All of us are blargs”, H says “Exactly one of us
is a nurt.” What are G, H, and J?
Problem 2. In Frienemytown, every pair of people is either friends with each other, or enemies
with each other (there is no in between option). Prove that if there are at least six people at a party
in Frienemytown, there is either a group of three pairwise friends OR of three pairwise enemies.
(A group of people are pairwise friends if each pair of people in the group are friends; a group of
people are pairwise enemies if each pair of people in the group are enemies. So George, Paul, and
John and pairwise friends if George and Paul are friends, Paul and George are friends, and John
and Paul are friends. However – if Paul is enemies with Ringo, then George, Paul, and Ringo are
neither pairwise friends nor pairwise enemies.)
Problem 3. Answer each of tthe following.
(a) Show that (∀ (x ∈ A) P (x)) ∨ (∀ (x ∈ A) Q (x)) is not equivalent to ∀(x ∈ A)(P(x) ∨ Q(x)).
On the other hand, show that ∀(x ∈ A)P(x) ∨ ∀(x ∈ A)Q(x) is logically equivalent to
∀(x ∈ A)∀(y ∈ A)(P(x) ∨ Q(y)). (Throughout, A can be any set)
(b) Show that (∀ (x ∈ A) P (x)) ∨ (∀ (x ∈ A) Q (x)) implies ∀(x ∈ A)∃(y ∈ A)(P(x) ∨ Q(y)). (Here,
you may assume that A is non-empty.)