Problems

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?

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 $2^n$.

Between two mirrors $AB$ and $AC$, forming a sharp angle two points $D$ and $E$ are located. In what direction should one shine a ray of light from the point $D$ in such a way that it would reflect off both mirrors and hit the point $E$?
If a ray of light comes towards a surface under a certain angle, it is reflected with the same angle as on the picture.