Types of Set and some commonly used Sets (Empty Set, Finite Set, Infinite Set, Singleton Set, Equal Sets, Equivalent Sets, Subsets, and Power Set)

Types of Sets and some commonly used Sets (Empty Set, Finite Set, Infinite Set, Singleton Set, Equal Sets, Equivalent Sets, Subsets, and Power Set)

Types of Set

There are three types of set

1. Empty Set/Null Set/Void Set
2. Finite Set
3. Infinite Set
   
   

Empty Set

• Empty means having no element
• A null set is a set that doesn't contain any elements.
• An empty set is represented by the Greek letter “$\ phi$” which is called “phi” or represented by curly braces “$\{\}$”.

Example:
$A$ $=\{x|x$ is a man with $200$ $\text{ feet height}\}$
$B$ $=\{x|xεN$  $˄$ $2<x<3\}$
$C$ $=$ The set of days starts with the letter “e”.

 

Finite Set

• Such a set whose number of elements can be counted.
• Such a set whose number of elements is limited
• We can count all the members.

Example:
$A=$ The set of the population of Pakistan.
$B=$ The set of hair on your head.
$C=$ $\{1,$ $2,$ $3,…$ $1000\}$

 

Infinite Set

• Such a set whose number of elements cannot be counted.
• We cannot count the number of elements of the infinite set.

Example:
$i)$    The set of rational numbers between $2$ and $3$.
$ii)$   The set of natural numbers.
$iii)$   $\{x|xεR$  $˄$  $0\le x$ $\le 1\}$

 

Some Important Sets

Singleton Set

• A set having only one element is called a singleton set.

Example:
$\{a+b\}$ ,  $\{0\}$ ,  $\{\frac{1}{2}\}$ ,  $\{x^2$ $+2x$ $+5\}$ ,  $\{x^2\}$

Equal Sets

• Two sets are considered equal if they have exactly the same elements.
• They are represented as $A=B$

Example:
$A=$ $\{2,$ $3,$ $5,$ $7\}$  and  $B$ $=\{3,$ $7,$ $5,$ $2\}$
Hence,   $A=B$

Equivalent Sets

• Let two sets be set "$A$" and set "$B$" is said to be equivalent sets if they have the same number of elements.
  i.e.  $n(A)$ $=n(B)$

• Equivalent sets are represented by the symbol "$\sim $".

Example:

$A$ $=\{1,$ $2,$ $3\}$  and  $B$ $=\{a,$ $b,$ $c\}$

then $A\sim B$ (equivalent sets)

Subsets

• A set "$A$" is said to be a subset of a set "$B$" if every element of a set "$A$" is also an element of a set "$B$".
• They are represented as $A⊆B$.

Example:

$A=$ $\{1,$ $2,$ $3\}$    and    $B=$ $\{1,$ $2,$ $3,$ $4,$ $5\}$    $A⊆B$

Note:  $1.$ Null set, $ɸ$ may be a subset of every set.
            $2.$ Every set may be a subset of itself.

There are two types of a subset

1. Proper Subset
2. Improper Subset

Proper Subset

• If $A\subseteq B$ and $A\ne B$ then $A\subset B$.
• Set "$A$" is said to be a proper subset of a set "$B$", If set "$A$"  is a subset of set "$B$" and if there exists at least one element in "$B$" that is not in "$A$".
• It is denoted as $A\subset B$. (read as "$A$" is a subset of set "$B$")
  

Example:

let $A=$ $\{1,$ $2,$ $3\}$ and $B$ $=\{1,$ $2,$ $3,$ $4,$ $5\}$ then $A\subset B$.

Remember:

1. An empty set is a proper subset of every non-empty set.
2. All the subsets of a set are proper subsets except itself.
3. There is only a proper subset of a singleton set.

Improper Subset

• If $A\subseteq B$ and $B\subseteq A$ then, $A$ and $B$ are said to be improper subsets of each other.
• If $A$ and $B$ are two subsets, set $A$ is a subset of set $B$. and $B$ is also a subset of set $A$ then $A$ and $B$ are improper subsets of each other.
• It is denoted as $A\subseteq B$ or $B\subseteq A$.

Note:  Every set is an improper subset of a set itself.

Power Set

• The set of all possible subsets of a set $A$ is called the power set of set $A$.
• It is denoted as $P(A)$.
• The total number of subsets of a finite set containing $n$ elements is $2^n$.
• The number of proper subsets of a set having $n$ elements is $2^n-1$.

Example:

$i)$  The number of non-empty subsets of a set $\{1,$ $2,$ $3,$ $4\}$ is.

$n(A)=4$

Non-empty subsets $2^{n-1}$ $=2^{4-1}$ $=16-1$ $=15$

$ii)$  If $A=\{1,$ $2,$ $3\}$ then find the total possible subsets.

$n(A)=3$,   $P(A)$ $=2^n$ $=2^3$ $=8$

$iii)$   If  $A=\phi$ then find $P(A)$.

$n(A)=0$,   $P(A)$ $=2^n$ $=2^0$ $=1$