Definition: A set is a well-defined collection of distinct objects.

These objects are called the elements or members of the set.

Sets are denoted by capital letters like, A, B, C, ......., X, Y, Z.

Its elements are written as,

a, b, c, .......

1, 2, 3, ....

Ram, Shyam, Hari,.......

Sita, Gita, Radha, ....

Apple, Mango, Cherry, .....

Inside the curly braces.

A = {a, e, i, o, u} $\rightarrow$ A set of vowels

B = {2, 4, 6, 8, 10} $\rightarrow$ A set of first five even numbers.

Notes:

Well-defined: This means there must be a clear criterion to determine whether any given object belongs to the collection or not. For example, "the collection of tall people" is not a set because "tall" is subjective. However, "the collection of people over 6 feet tall" is a set.
Distinct: Each object in a set must be unique. If an element is listed more than once, it is still considered a single member of that set.
Order-independent: The order in which elements are listed does not change the set. $\{1, 2, 3\}$ is the same as $\{3, 1, 2\}$.

Type of Sets

Activity:-

The following sets are given,

A = { seven feet tall boy in your class }
B = { 8th colour in a rainbow }

Observation:

Set A and B don't contain any element or we can say there are no elements in sets A and B.

Conclusion:

These sets are called null or empty set.

Definition:

The set with no element in it is called empty set or null set.

It is denoted as Ø (phi), { }.

Note:-

Let A = {0},
Set A is not a empty set as it contain a element "0" in it.

Activity:-

There are two lists of sets.

List - 1: A = { vowels of english alphabet }
List - 2: B = { even numbers }

Observation:-

A = { a, e, i, o, u }
B = { 2, 4, 6, 8, 10, 12, ...... }
We can count elements of set A.
We cannot count elements of set B as there is no end value.

Conclusion:-

Set A from (List-1) has fixed elements in it, so it is finite set.
Set B from (List-2) don't have fixed elements in it, so it is infinite set.

Definition

The set having fixed number of elements in it is called finite set.
The set which don't have fixed number of elements in it is called infinite set.

Activity:-

Here are two groups of sets.

Group '1'	Group '2'
Set X = { 1, 2, 3, 4 } Set Y = { 3, 1, 2, 4 }	Set P = { apple, banana, Mango } Set Q = { Red, blue, Green }

Observation:-

Sets of Group '1' (X and Y) have same elements in them.
Sets of Group '2' (P and Q) have different elements in them.
Sets of group '2' (P and Q) have equal number of elements in them.

Conclusion:-

Sets of Group '1' (X and Y) are equal sets.
Sets of Group '2' (P and Q) are equivalent sets.

Definition:-

Two non-empty sets A and B are said to be equal sets if they have same elements.

Equal sets are denoted as A = B.

Two non-empty sets P and Q are said to be equivalent sets if they have same number of elements.

Equivalent sets are denoted as P ~ Q.

Activity - 1

$S_1$ = { set of girls of class - 7 }
$S_2$ = { set of boys of class - 7 }
$S_3$ = { set of students of class - 7 }

Observation:

$S_3$ contains all elements of sets $S_1$ and $S_2$.

Activity - 2

$S_1$ = { 2, 4, 6, 8 }
$S_2$ = { 1, 3, 6, 9 }
$S_3$ = { 1, 5, 7, 10 }
$S_4$ = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

Observation:

Set $S_4$ contains all elements of sets $S_1$, $S_2$, $S_3$.

Conclusion:-

(Activity-1): Set $S_3$ is the universal set of sets $S_1$ and $S_2$.

(Activity-2): Set $S_4$ is the universal set of sets $S_1$, $S_2$, $S_3$.

Definition:-

A universal set is the master set which contains all elements of sets under consideration.

Generally, Universal set is written using letter $\textbf{U}$.

SUBSETS

Let us consider a set
$A = \{1, 2, 3\}$

Now let us create different sets using elements of set $A$.

$\bullet$ $S_1 = \{1\}$
$\bullet$ $S_2 = \{2\}$
$\bullet$ $S_3 = \{3\}$
$\bullet$ $S_4 = \{1, 2\}$
$\bullet$ $S_5 = \{2, 3\}$
$\bullet$ $S_6 = \{1, 3\}$
$\bullet$ $S_7 = \{ \}$ [Empty set is like air, present everywhere]
$\bullet$ $S_8 = \{1, 2, 3\}$

Observation:-
$\bullet$ Sets $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ are made by using elements of set $A$ only.
$\bullet$ Sets $S_1, S_2, S_3, S_4, S_5, S_6, S_7$ are not equal to set $A$.
[Equal and equivalent sets are studied in previous class]
$\bullet$ Set $S_8$ is equal to set $A$.

Conclusion:
$\bullet$ Sets $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ are all SUBSETS of set $A$.
$\bullet$ Sets $S_1, S_2, S_3, S_4, S_5, S_6, S_7$ are PROPER SUBSETS.
$\bullet$ Set $S_8$ is IMPROPER SUBSETS

Definition:-
SUBSETS
A set within a set (सेट भित्रको सेट) is called SUBSET.

PROPER SUBSET
Let $A$ and $B$ be any two sets.
Set $B$ is said to be proper subset of $A$ if:
$\bullet$ $B$ contains elements of $A$ only
$\bullet$ $B$ is not equal to $A$.

It is written as: $B \subset A$

IMPROPER SUBSET
Let $A$ and $B$ be any two sets.
Set $B$ is said to be improper subset of $A$ if:
$\bullet$ $A = B$

It is denoted as: $B \subseteq A$

Keep in Mind
$\bullet$ Empty / Null set is a subset of every set.
$\bullet$ Each set will have only one improper subset and rest of the subset are proper subsets.

Formula To calculate subsets:

Sets	No of elements ($n$)	Subsets	No. of subsets
$\{a\}$	1	$S_1 = \{a\}, S_2 = \{\}$	$2 = 2^1$
$\{a, b\}$	2	$S_1 = \{a\}, S_2 = \{b\}, S_3 = \{a, b\}, S_4 = \{\}$	$4 = 2^2$
$\{a, b, c\}$	3	$S_1 = \{a\}, S_2 = \{b\}, S_3 = \{c\},$ $S_4 = \{a, b\}, S_5 = \{b, c\},$ $S_6 = \{a, c\}, S_7 = \{\}, S_8 = \{a, b, c\}$	$8 = 2^3$
$----$	$----$	$----------$	$----$
$\{a, b, c, \dots\}$	$n$	$------$	$= 2^n$

$\therefore$ No of total / possible subsets = $2^n$

or, no of (proper + improper) subsets $= 2^n$
or, no of proper subsets + 1 improper subset $= 2^n$
or, no of proper subsets = $2^n - 1$