Problems

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

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.

Let $A=\{1,2,3\}$ and $B=\{2,4\}$ be two sets containing natural numbers. Find the sets: $A\cup B$, $A\cap B$, $A-B$, $B-A$.

Let $A=\{1,2,3,4,5\}$ and $B=\{2,4,5,7\}$ be two sets containing natural numbers. Find the sets: $A\cup B$, $A\cap B$, $A-B$, $B-A$.

Given three sets $A,B,C$. Prove that if we take a union $A\cup B$ and intersect it with the set $C$, we will get the same set as if we took a union of $A\cap C$ and $B\cap C$. Essentially, prove that $(A\cup B)\cap C=(A\cap C)\cup (B\cap C)$.

$A,B$ and $C$ are three sets. Prove that if we take an intersection $A\cap B$ and unite it with the set $C$, we will get the same set as if we took an intersection of two unions $A\cup C$ and $B\cup C$. Essentially, prove that $(A\cap B)\cup C=(A\cup C)\cap (B\cup C)$. Draw a Venn diagram for the set $(A\cap B)\cup C$.

Let $A,B$ and $C$ be three sets. Prove that if we take an intersection $A\cap B$ and intersect it with the set $C$, we will get the same set as if we took an intersection of $A$ with $B\cap C$. Essentially, prove that it does not matter where to put the brackets in $(A\cap B)\cap C=A\cap (B\cap C)$. Draw a Venn diagram for the set $A\cap B\cap C$.
Prove the same for the union $(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C$.

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?