Sets
Digital Handwritten Lesson
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 $\{3, 1, 2\}$.
Relation Between The Sets:
Ram formed a set as R = {2, 4, 6, 8, 10, 12}. His set is a set of first 6 even numbers.
Sita formed a set S = {3, 6, 9, 12, 15}. Her set is a set of first 5 multiples of $3$.
Observation:
- Set R and set S have $6$ and $12$ common.
- Set R and set S both contain elements $6$ and $12$.
- $6$ and $12$ are present on both sets R and S.
Conclusion: Set R and set S are Overlapping sets.
Definition:
Two sets are said to be overlapping sets if they have atleast (थोरैमा पनि एउटा) one element common.
Teacher asked Jay and Biru to write sets using 4 elements:
Jay wrote J = {1, 3, 5, 7}. His set is set of first 4 prime numbers.
Biru wrote B = {2, 4, 6, 8}. His set is set of first 4 even numbers.
Observation:
- Set J and B have no element common.
- Set J and B have distinct elements.
Conclusion: Set J and B are called Disjoint sets.
Definition:
Two sets are said to be disjoint sets if no element is common between the sets.
VENN DIAGRAM
History: Venn diagram was developed by John Venn in 1880s.
Purpose: To show the relationships between different sets.
Key components:
Definition:
Venn diagram is a tool (साधन) used to show relationship between the sets.
Note: While drawing a Venn diagram, students must be aware of few important points:
i) Students must be aware of presence of universal set. If universal set is given then rectangle must be drawn (like figure 5 or 6) other wise no need (like 3 or 4).
ii) Students must identify whether the sets are Overlapping or disjoints.
a) If overlapping the figure 3 (from above) must be drawn and have to fill the common elements first in overlapped region.
b) If disjoint the figure 4 (from above) must be drawn.
Example: 1 (Overlapping Sets) 
Let:
-
Set $A = \{2, 4, \boxed{6}, 8\}$
-
Set $B = \{3, \boxed{6}, 9, 12\}$
Venn Diagram Breakdown:
The Venn diagram shows two overlapping circles, labeled A and B.
-
Set A Only: This region contains the elements unique to A: $\{2, 4, 8\}$.
-
Overlapping Area: The number 6 is common to both sets, so it is placed here.
-
Set B Only: This region contains the elements unique to B: $\{3, 9, 12\}$.
Explanation:
$\{\text{Clearly, `6' is common, so sets are overlapping}\}$
The region where the circles intersect is explicitly labeled as the ``overlapping area."
Example 2: (Disjoint Sets) 
Let:
-
Set $A = \{1, 3, 5, 7\}$
-
Set $B = \{2, 4, 6, 8\}$
Venn Diagram Breakdown:
Because there are no common elements, this diagram shows two completely separate (non-touching) circles, labeled A and B.
-
Circle A: Contains the entire set of elements $\{1, 3, 5, 7\}$.
-
Circle B: Contains the entire set of elements $\{2, 4, 6, 8\}$.
Explanation:
$\text{Clearly, no element is common, so sets are disjoint.}$
Example 3: (With Universal Set)
Let:
-
Universal Set ($U$): $$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$
-
Set $A$ (Odd Numbers):
$$A = \{x : x \text{ is an odd number and } x \in U\}$$
$$A = \{1, \boxed{3}, \boxed{5}, \boxed{7}, 9\}$$
-
Set $B$ (Prime Numbers):
$$B = \{x : x \text{ is a prime number and } x \in U\}$$
$$B = \{2, \boxed{3}, \boxed{5}, \boxed{7} \}$$
$$\boxed{\text{Clearly}, \textbf{A} \text{and} \textbf{B} \text{are \textcolor{red}{Overlapping sets}, because elements `3', `5', `7' are common or present in both sets.}}$$
$\textbf{Venn Diagram}$:
$$\boxed{\text{Remember! Universal set is given so outer rectangle must be drawn.}}$$
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 grade 7]
$\bullet$ 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 = $2^n - 1$
Course material curated by Mr. Nripendraswar Acharya