∆ Definition 1. A set is a collection of well defined and distinct objects which are called elements. A possible way to define a set is to list its elements inside braces. Notation: "$x \in A$" means element $x$ belongs to set $A$.

Problem 1. How many elements are there in the following sets: a) $\{1\}$, $\{1,2,3\}$, $\{\text{Ann}\}$; b) $\{\{1\}\}$; c) $\{1, \{2, 3\}\}$; d) letters in the word "crocodile"; e) names of the Fields medalists? Send a solution

∆ Definition 2. Two sets $A$ and $B$ are called equal, if each element of $A$ belongs to $B$ and each element of $B$ belongs to $A$. Notation: $A=B$.

∆ Definition 3. Set $A$ is called a subset of set $B$, if each element of $A$ belongs to $B$. Notation: $A \subset B$. One way to define a subset is to define a property that all of the elements in the subset have in common. Notation: {$x \in A$ | $x$ has property $y$}.

Problem 2. a) Let $A$ be the set of one-digit natural (positive integer) numbers. Define its subset ${2,4,6,8}$ using the notation from the definition above. b) Let $A$ be the set of all cities in the USA. List all the elements of the following subset {$x \in A$ | the population of $x$ is more than 1 000 000 as of 01/01/2015} Send a solution

Problem 3. For each two of the following sets show if one is a subset of the other: \begin{equation*} \{1\}, \{1,2\}, \{1,2,3\},  \{\{1\},2,3\}, \{\{1,2\},3\},  \{3,2,1\}, \{\{2,1\}\}. \end{equation*} Send a solution

Problem 4. Show that a set $A$ is a subset of set $B$ if and only if any element that doesn't belong to $A$ doesn't belong to $B$. Send a solution

Problem 5. Show that for arbitrary sets $A$, $B$ and $C$ the following is true: a) $A \subset A$, b) if $A \subset B$ and $B \subset C$, then $A \subset C$, c) $A=B \Leftrightarrow A \subset B$ and $B \subset A$. Send a solution

∆ Definition 4. A set is called empty if it contains no elements. Notation: $A = \varnothing$.

Problem 6. a) Show that the empty set is a subset of any set. b) Show that the empty set is unique. Send a solution

Problem 7. How many elements do the following sets have: \begin{equation*} \varnothing,  \{1\}, \{1,2\}, \{1,2,3\}, \\   \{\{1\},2,3\}, \{\{1,2\},3\},  \{\varnothing\}, \{\{2,1\}\}? \end{equation*} Send a solution

Problem 8. а) For each of the sets from the previous problem write down all their subsets. b) How many subsets does a set of one element have? of two elements? three? Send a solution

Problem 9. Is it true that the set of flying crocodiles is a subset of the set of Evarist users? Send a solution

Problem 10. Is it possible for a set to have exactly: a) $0$, b) $7$, c) $16$ subsets? Send a solution

∆ Definition 5. The union of sets $A$ and $B$ is a set consisting of all elements $x$ such that $x \in A$ or $x \in B$. Notation: $A \cup B$. The intersection of sets $A$ and $B$ is a set consisting of all elements $x$ such that $x \in A$ and $x \in B$. Notation: $A \cap B$. The difference of sets $A$ and $B$ is a set consisting of all elements $x$ such that $x \in A$ and $x \notin B$. Notation: $A \setminus B$.

Problem 11. Consider the following sets $A=\{1,3,7,137\}$, $B=\{3,7,23\}$, $C=\{0,1,3,23\}$, $D=\{0,7,23,2004\}$. Find the following sets: 1) $A\cup B;$   2) $A\cap B$;   3) $(A\cap B)\cup D$;   4) $C\cap (D\cap B)$; 5) $(A\cup B)\cap (C\cup D)$; 6) $(A\cup (B\cap C))\cap D$;   7) $(C\cap A)\cup ((A\cup (C\cap D))\cap B)$; 8) $(A\cup B)\setminus (C\cap D)$;  9) $A\setminus (B\setminus(C\setminus D))$;  10) $((A\setminus (B\cup D))\setminus C)\cup B$. Send a solution

Problem 12. Let $A$ be the set of even numbers and $B$ — the set of numbers divisible by three. What is $A \cap B$? Send a solution

Problem 13. Prove that for any sets $A$, $B$, $C$ the following is true 1) $A\cup B=B\cup A$;  $A\cap B=B\cap A$; 2) $A\cup (B\cup C)=(A\cup B)\cup C$;  $A\cap (B\cap C)=(A\cap B)\cap C$; 3) $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$;  $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$ 4) $A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)$;  $A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C)$. Send a solution

Problem 14. Is it true that for any sets $A$, $B$, $C$ 1) $A\cap\varnothing=\varnothing$;  $A\cup\varnothing=A$;   2) $A\cup A=A$;  $A\cap A=A$; 3) $A\cap B=A\Leftrightarrow A\subset B$; 4) $(A\setminus B)\cup B=A$;   5) $A\setminus (A\setminus B)=A\cap B$; 6) $A\setminus (B\setminus C)=(A\setminus B)\cup (A\cap C)$;   7) $(A\setminus B)\cup (B\setminus A)=A\cup B$? Send a solution

Problem 15. а) Inside a figure of area $6$ there are three polygons of area at least $3$ each. Show that the area of the intersection of two of these polygons is at least $1$. b*) Inside a figure of area $4$ there are seven polygons of area at least $1$ each. Show that the area of the intersection of two of these polygons is at least $1/7$. Send a solution

Problem 16. a*) Is it possible to express the intersection of two sets using only union and difference? b) Is it possible to express the difference of two sets using only union and intersection? Send a solution