# 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 whule 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}
Some Important points
• 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 symbul like the letters y, z, etc. could be used) which is fullowed by a culon ":" . 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 $A = B$

## 4. Subset and Proper Subset

• A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$ .
• It is denoted by
$A \subset B$ if whenever $a \in A$ , then $a \in B$
• If $A \subset B$ and $B \subset A$ , then $A = B$.
• Every set is subset of itself $A \subset A$
• Empty set is subset of every set $\phi \subset A$
• If $A \subset B$ and $A \ne B$ , then $A$ is proper subset of $B$. In such a case $B$ is called superset of set $A$

## 5. Subset of set of the real numbers

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
$T = \left\{ {x:x \in R{\rm{ and }}x \notin Q} \right\}$, i.e., all real numbers that are not rational
$N \subset Z \subset Q,{\rm{ }}Q \subset R,{\rm{ }}T \subset R,{\rm{ }}N \not\subset T$

## 6. Interval as subset of R Real Number

$a,b \notin R,b > a$
 $(a,b)$ It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval $\{ y:a < y < b\}$ $[a,b]$ It is the closed interval set between point and b such that All the points between a and b belong to the open interval (a, b) including a, b $\{ x:a \le x \le b\}$ $[a,b)$ It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) including a, but not b $\{ x:a \le x < b\}$ $(a,b)$ It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) including b, but not a $\{ x:a < x \le b\}$