Problems

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?

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.

Today you saw two infinitely long buses with seats numbered as $1,2,3,...$ carrying infinitely many guests each arriving at the full hotel. How do you accommodate everyone?

Now there are finitely many infinitely long buses with seats numbered as $1,2,3,...$ carrying infinitely many guests each arriving at the full hotel. Now what do you do?