 # Sets Theory| Types of Sets

## 1. Introduction

What is Sets?
Simply put, it's a collection of objects
Examples
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}

## 2. Methods of representing a set

• For sets, we simply put each element, separated by a comma, and then put some curly brackets around the whole thing.
• Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc
• The elements of a set are represented by small letters a, b, c, x, y, z, etc
• When we say an element a is in a set A, we use the symbul $\in$ to show it.
And if something is not in a set use $\notin$
Example:
In a set of even number $E$, $2 \in E$ but $3 \notin E$

Two Methods are used to represent Sets

### (a) Roster forms

• In a Roster forms, all the elements in the set is listed.
Example
Set of Vowel ={ a,e,i,o,u}
• In roster form, the order in which the elements are listed is immaterial
• while writing the set in roster form an element is not generally repeated

### (b) Set Builder Form

• In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.For example, in the set $\left\{ {2,4,6,8} \right\}$, all the elements possess a common property, namely, each of them is a even number less than 10. Denoting this set by $N$, we write
$N = \left\{ {x:x{\rm{\; is \; a \; even \; number \; less \;10}}} \right\}$
• We describe the element of the set by using a symbul x (any other symbol like the letters y, z, etc. could be used) which is followed by a colon ":" . After the sign of culon, we write the characteristic property possessed by the elements of the set and then enclose the whule description within braces

## 3. Types of sets

### (a) Empty set

• A set which does not contain any element is called the empty set or the null set or the void set
• It is denoted by $\phi$ or $\left\{ {} \right\}$
• It is a set with no elements
• Examples of empty sets is
$D = \left\{ {x:{x^2} = 9,x{\rm{\; is \; even}}} \right\}$
Here D is the empty set, because the equation ${{x^2} = 9}$ is not satisfied by any even value of x

### (b) Finite or infinite set

• If $M$ is a set then $n\left( M \right)$ defines the number of distinct elements in the set M.
• If $n\left( M \right)$ is zero or finite ,then $M$ is a finite set
• If $n\left( M \right)$ is infinite then $M$ is a infinite set

### (c) Equal sets

• Two sets are said to be equal if they have same members in them.
• For $A$ and $B$ to be equal, every member of $A$ should be present in set $B$ and every member of $B$ to be present in set $A$
• It is denoted by equality sign $A = B$

### (d) singleton sets

• A sets are said to be singleton if has just one element in it
• A ={a } is a singleton set

## Solved Example

Question 1
Write the set {$\frac {1}{2}, \frac {2}{3}, \frac {3}{4}, \frac {4}{5}, \frac {5}{6}$} in the set-builder form.
Solution
It can be observed that each member in the given set has the numerator one less than the denominator. Also, the numerator begin from 1 and do not exceed 5. Hence, in the set-builder form the given set is
{x: $x= \frac {n}{n+1}$,where n is a natural number and 1 ≤ n ≤ 5}

Question 2
State which of the following sets are finite or infinite :
(i) {x : x ∈ N and (x - 4) (x - 5)(x- 6) = 0}
(ii) {x : x ∈ N and x3 = 8}
(iii) {x : x ∈ N and 2x - 5 = 0}
(iv) {x : x ∈ N and x is prime}
(v) {x : x ∈ N and x is even}
Solution
(i) Given set = {4, 5,6}. Therefore, it is finite.
(ii) Given set = {2}. Therefore, it is finite.
(iii) Given set = f. Therefore, it is finite.
(iv) The given set is the set of all prime numbers and since set of prime numbers is infinite. Therefore the given set is infinite
(v) Since there are infinite number of even numbers, Therefore, the given set is infinite.

### Quiz Time

Question 1 Two sets are given $A = {x : x - 5 = 0 }$ and $B = {x : x \; is \; an \; integral \; positive \; root \; of \; the \; equation x^2 - x -20 = 0}$.
A) $A \neq B$
B) $A = B$
C) $A= \phi$
D) $B= \phi$
Question 2 which of these is true ?
A) $28 \in {y | the \; sum \;of \; all \;the \;positive \;factors \; of \;y \;is \; 2y}$
B) $37 \in {x | x \; has \; exactly \; two \; positive \; factors}$
C) $4 \in {x|x^2 -9x +14=0}$
D) None of the above
Question 3 which of these is a finite set
A) Set of lines which are parallel to the y-axis
B) set of odd prime numbers
C) set of positive integers greater than 35
D) set of letter in english alphabet
Question 4 Interval [-1, 2) can be written in setbuilder form as
A) ${x : x \in R, -1 < x < 2}$
B) ${x : x \in R, -1 < x \leq 2}$
C) ${x : x \in R, -1 \leq x \leq 2}$
D) ${x : x \in R, -1 \leq x < 2}$
Question 5 A = { 1,2,3,4,5} and B ={2,3,4,5} then
A) $B \subset A$
B) $A \subset B$
C)$A = B$
D)none of the above
Question 6$A = {x | x^2 = x, x \in R}$ can be written in roster form as
A) A={1}
B) A={-1}
C) A={0,1}
D) A={-1,0,1}