Problems

For three sets \(A,B,C\) prove that \(A - (B\cup C) = (A-B)\cap (A-C)\). Draw a Venn diagram for this set.

For three sets \(A,B,C\) prove that \(A - (B\cap C) = (A-B)\cup (A-C)\). Draw a Venn diagram for this set.

How many subsets of \(\{1, 2, . . . , n\}\) are there of even size?

In how many ways can \(\{1, . . . , n\}\) be written as the union of two sets? Here, for example, \(\{1, 2, 3, 4\}\cup\{4, 5\}\) and \(\{4, 5\}\cup\{1, 2, 3, 4\}\) count as the same way of writing \(\{1, 2, 3, 4, 5\}\) as a union.

Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).

Consider a set of natural numbers \(A\), consisting of all numbers divisible by \(6\), let \(B\) be the set of all natural numbers divisible by \(8\), and \(C\) be the set of all natural numbers divisible by \(12\). Describe the sets \(A\cup B\), \(A\cup B\cup C\), \(A\cap B\cap C\), \(A-(B\cap C)\).

Prove that the set of all finite subsets of natural numbers \(\mathbb{N}\) is countable. Then prove that the set of all subsets of natural numbers is not countable.

There are infinitely many couples at a party. Each pair is separated to form two queues of people, where each person is standing next to their partner. Suppose the queue on the left has the property that every nonempty collection of people has a person (from the collection) standing in front of everyone else from that collection. A jester comes into the room and joins the right queue at the back after the two queues are formed.

Each person in the right queue would like to shake hand with a person in the left queue. However, no two of them would like to shake hand with the same person in the left queue. If \(p\) is standing behind \(q\) in the right queue, \(p\) will only shake hand with someone standing behind \(q\)’s handshake partner. Show that it is impossible to shake hands without leaving out someone from the left queue.

King Hattius has three prisoners and gives them the following puzzle. He will put a randomly coloured hat on each of their heads: red, blue or green. He’ll then give them \(10\) seconds for them to each guess their own hat’s colour at the same time.

However! Each prisoner can only see the other two prisoners’ hats, not their own. There are no mirrors in the prison, and they are not allowed to take off their hat, nor talk, mouth, use sign-language, or otherwise communicate with the other two prisoners during those ten seconds.

Hattius tells them that he’ll release them all if at least one correctly guesses their hat’s colour. He gives them an hour to come up with a strategy - what should their strategy be?

Two aliens want to abduct two humans, but aren’t paying attention, so instead run after pigs. They’re all on squares of a \(3\times6\) rectangle, as seen below. On the first move, the aliens move one square horizontally or vertically. Then on the second move, the pigs move horizontally or vertically. The third move is for the aliens, the fourth move is for the pigs, and so on. If an alien lands on a square with a pig on it, then they’ve succeeded. Show that no matter what the pigs do, they’re doomed.