Home / Set Theory / Sets

**Sets**

**What are sets?**

To name different group of objects in our daily life we use different words, like a group of students, pile of stones, or herd of fold. In mathematic we use the word set. So in other words,

**‘Sets are collection of things that are grouped together based on a common property.’**

The objects that are included on the set are called the elements of a set.

The sets are usually defined with capital letters and its elements with small letters.

**Examples**

The set of all the odd numbers less than 20.

**Solution:** We define the set with the capital letter O

Then write all its elements

**O** = {**1**, **3**, **5**, **7**, **9**, **11**, **13**, **15**, **17,** **19**}

For the number that is included on the set we write:

For the number that is not included on the set we write:

**How to present a set?**

**
Method 1: **Tabular Form

It’s a form of presenting a set while writing all the elements of the set in curly brackets.

**Example 1: **The set of all the vowels.

**Solution:** **E** = {**a**, **e**, **o**, **u**, **i**}

**Example 2: **The set of natural numbers less than 10 and more than 2.

**Solution:** **N** = {**3**, **4**, **5**, **6**, **7**, **8**, **9**}

**Method 2: **Descriptive Form

It’s a form of presenting the set by writing the common characteristic of all her elements.

**Example:**

**E = **Set of all the vowels

**N = **Set of all the natural numbers that are more than 2 and less than 10

**O **= Set of all odd numbers less than 20

**Method 3: **Set builder form

It’s a form when you write the common characteristic of all the elements on a symbolic way.

**Example:**

**A** = {x: x N, **2 < x < 10** } N – natural numbers

**B** = {x: x O, **x < 2****0**} O – odd numbers

**Method 4: **Venn Diagram

It’s a form when you draw a closed line and inside you write all the elements of the set.

**The equality of sets**

Two sets are called equal if they are made from the same elements. With other words every element of one set is an element of the other set and conversely.

**Example: **The set **A** = {**1**, **2**, **3**, **4**, **5**, **6**} and the set **B** = {**1**, **2**, **3**, **4**, **5**, **6**} are equal because they have the same elements regardless the order of the elements.

**The subset of a set**

The set B is called a subset of the set A if every element of the set B is included on the set A. In short we write **B A**

**Example: **Let’s have the set **A** = {**1**, **2**, **3**, **4**, **5**, **6**} and the set B = {**1**, **3**, **5**}

We see that every element of the set B is an element of the set A. So we write **B A.**

{**1**, **3**, **5**} **A** = {**1**, **2**, **3**, **4**, **5**, **6**}

**Types of sets**

**Type 1:** Empty set

An empty set is a set which does not contain any element. It is written with the symbol Ø

**Example: ****N = **the set of all the natural numbers between -1 and 0

As u can see there is no natural number between -1 and 0 In other words our set is an empty set **N = ****Ø**

**Type 2: **Finite set

A finite set is a set which has a finite number of elements and u can list them even when they are a lot.

**Example: **E = the set of all the vowels of the alphabet

**A** = {**1**, **2**, **3**, **4**, **5**, **6**}

**A** = {x: x N, **2 < x < 10** }

**Type 3:** Infinite set

An infinite set is a set which has an infinite number of elements that can’t be listed even if we want to.

**Example: **N = the set of all the natural numbers

**E** = the set of all the even numbers

**A** = {x: x Q, **x > 2** } Q-rational number

**Type 4: **Universal set

Is a set that contains all the possible elements that you would consider for a set in a particular problem. The symbol used is U.

**What is the complement of a set?**

The complement of a set A is the set of all things that are in U but not in the set A. The symbol A´ is used to denote the complement of a set A.

**For example:**

If U={1,2,3,4,5,6,7,8,9,10} and A={2,4,6}

Then the complement of A would be A´= {1,3,5,7,8,9,10}

** **

**Intersection of sets**

Intersection of two sets is a set that contains all the common elements of two sets.

The symbol we use to write the intersection of two sets is

The intersection of two sets A and B

** **

**Union of the sets**

Union of two sets is a set that contains all the elements of both of the sets.

The symbol we use to write the union of two sets is

The union of two sets A and B

** **

**Example 1: **Find the union and the intersection of the given sets.

**A** = {1, 2, 3, 4, 5, 6**, **7**, **8}

**B** = {2, 4, 8, 10, 12}

** Solution: **The intersection of these two sets is:

A B = {**2**, **4**, **8**,}

**The union of these two sets is:**

A B = {**1**, **2**, **3**, **4**, **5**, **6**, **7**, **8**, **10**, **12**}

**Example:** If C= {4,8,12,16,20,24} and D={5,8,20,24,28}

**a)** Lists the sets C∪D and C∩D.

**b)** Is it true that D⊂C ?

**Solution**

**a) **C∪D is the set of all members of C or of D or of both

**C∪D= {4, 5, 8, 12, 16, 20, 24, 28}**

C∩D is set of all elements that apper in both C and D

**C∩D={8,20,24}**

**b)** Notice that 5∈D but 5∉C. So it is not true that every member of D is also a member of C. So, D is not a subset of C.