## Motivation

Take a look at these objects; what are their similarities and differences? Can you classify them by some criterion?

You can notice that objects denoted by numbers $1, 3$ and $5$ are plane figures, i.e. two – dimensional (2D) shapes, while objects denoted by numbers $2, 4$ and $6$ are solid figures, i.e. three – dimensional (3D) shapes. Therefore, we have two **collections**: collection of plane figures and collection of solid figures. Obviously, figures in each of the two collections have **the same property**. We can tell that collection of objects with some property in common is called **a set**.

Some more examples of sets are: the set of people younger than $25$ years, the set of flowers, the set of four seasons, the set of string instruments and so on.

A set which has no elements is called an **e****mpty set** or **null set**. We denote it by $\emptyset$.

## Set notation

Members of set are called **elements**. Notations for mentioned sets are: **A =** **{$1, 3, 5$}** and **B = ****{$2, 4, 6$}** or **A = {triangle, quadrilateral, circle}** and **B = {parallelepiped, pentagonal prism, triangular pyramid}**. Therefore, we denote sets by a single capital letter, simply list the elements, separate them by comma and put curly brackets.

The notation for set of plants is: {tulip, sunflower, rose, … }; we use the three dots when the set has a large number of elements or when it is **infinite**. Sets which aren’t infinite are called **finite sets**.

Size of the set $A$ is called **cardinality number** and it is denoted by $|A|$. Cardinalities of mentioned sets $A$ and $B$ are: $|A|=3$ and $|B|=3$. Cardinality of empty set is $0$.

Other way of representing a set is describing a property that its element must satisfy. For example, we denote set of all numbers which are divisible by $7$ as:

$\{$ x: x is divisible by 7 $\}$ or $\{$ x|x is divisible by 7 $\}$.

If $x$ is an element of a set $A$, we write: **$x \in A$** and say **”$x$ belongs to $A$”**.

If $x$ isn’t the element of a set $A$, we write: **$x \notin A$** and say **”$x$ doesn’t belong to $A$”**.

* Example 1: *Write a collection of vowels in English alphabet. How many elements does it have? Are letters $b, d$ and $o$ elements of that collection?

**Solution:**

$V= \{a, e, i, o, u\}$, it has $5$ elements.

$b \notin V$, $d \notin V$, $o \in V$.

## Properties of sets

**The order in which elements of a set are listed is not important.** For example, $\{a, e, i, o, u\} = \{u, o, a ,i, e\}$.

**If some elements are repeated, set is still the same.** For example, $\{3, 3, 7, 8\} = \{3, 7, 8\}$.

**Two sets are equal if and only if they have the same elements. **For example, $A$ = set of primary colors and $B = \{red, blue, yellow\}$ are equal.

## Subsets

A set $A$ is said to be a **subset** of the set $B$ if and only if every element of set $A$ is also the element of set $B$. We write:

**$A \subseteq B$**.

In that case, $B$ is the **superset** of set $A$.

**Empty set is a subset of every set.**

If $A$ is not the subset of $B$, we write: $A \not \subseteq B$.

**Universal set U** is a set that is superset of all sets.

* Example 2: *Is $A$ subset of $B$, where $A = \{$x: x is a whole number greater than $4$ $\}$ and $B = \{$x: x is even number $4<x<15$ $\}$?

**Solution:**

$A= \{$$5, 6, 7, 8, …$$\}$ and $B = \{$$6, 8, 10, 12, 14$$\}$

$A \not \subseteq B$, $B \subseteq A$

A set $A$ is said to be a **proper subset** of the set $B$ if and only if every element of set $A$ is also the element of set $B$, but there exists at least one element which is in $B$ and not in $A$. We write:

**$A \subset B$**.

If $A$ is not a proper subset of $B$, we write: $A \not \subset B$.

** Example 3: **Find proper subsets of set $A=\{$$a, b$$\}$.

**Solution:**

Proper subsets are: $\emptyset$, $\{$a$\}$, $\{$b$\}$.

**Note: $A=B \Leftrightarrow (A \subseteq B \wedge B \subseteq A)$**

## Power set

A **power set** of a set $S$, denoted by $\mathcal{P}(S)$, is a set of all the subsets of a set. If a set has $n$ elements, power set has $2^{n}$ elements.

* Example 4: *Find $\mathcal{P}(S)$, where $S=\{a, b\}$.

**Solution:**

$\mathcal{P}(S)= \{\emptyset, \{a\}, \{b\}, \{a, b\}\}$