Problems

Prove the triangle inequality: in any triangle $ABC$ the side $AB<AC+BC$.

In certain kingdom there are a lot of cities, it is known that all the distances between the cities are distinct. One morning one plane flew out of each city to the nearest city. Could it happen that in one city landed more than $5$ planes?

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?