Kushagra Agrawal submitted a mathematics project to Kamal Soni Sir. The project discusses sets and set operations including intersections and unions of sets. It provides examples of intersections and unions of sets and uses Venn diagrams to prove the distributive law of sets. The project contains an index listing the topics covered on each slide and cites references used at the end.

1. MATHEMATICS
PROJECT
Submitted by : Kushagra Agrawal
Submitted to : Kamal Soni Sir

2. Acknowledgement
The success and final outcome of this project required
a lot of guidance and assistance from many people and
I am extremely privileged to have got this all along the
completion of my project. All that I have done is only
due to such supervision and assistance and I would
not forget to thank them.
I respect and thank Kamal Soni Sir , for providing me
an opportunity to do the project work in The Sanskaar
Valley School and giving us all support and guidance
which made me complete the project duly. I
am extremely thankful to him for providing such a nice
support and guidance, although he had busy schedule.

3. INDEX
S. NO. TITLES SLIDE NO
1 What Are Sets? 4
2 Venn diagrams 5
3 Intersection of Sets 6
4 Union of Sets 7
5 Examples 8
6 Proving the Distributive Law of Sets 9

4. What are Sets?
A set is a collection of distinct objects. The objects can be called elements
or members of the set.
A set does not list an element more than once since an element is either a
member of the set or it is not.
The set can be defined by listing all its elements, separated by commas and
enclosed within braces. This is called the roster method.
Example : X = {a, b, 23, 43, 97}
Or where possible, by describing the elements. This is called the set-builder
notation.
Example : X = {z : z is a whole number, z < 2,00,42,020}

5. Venn Diagrams
A Venn Diagram is a pictorial representation of the relationships
between sets.
In a Venn diagram, the sets are represented by shapes; usually circles
or ovals. The elements of a set are labeled within the circle.
Example:
Boys
23
Girls
24

6. Intersection of Sets
The intersection of two sets has only the elements common
to both sets.
If an element is in just one set it is not part of the intersection.
It is denoted by X ∩ Y and is read ‘X intersection Y ’.
In a Venn diagram, the highlighted area represents the intersection
of the Sets :

7. Union Of sets
The union of two sets A and B is the set of elements, which are in
A or in B or in both.
It is denoted by A ∪ B and is read ‘A union B’
In a Venn diagram, all the elements inside the diagram represents
the union of the sets :

8. Examples
Intersection of Sets :
If X = {2, 5, 7, 8, 28, 13};
& Y = {3, 5, 13, 4, 9, 8};
X ∩ Y = {5, 8, 13};
Union of Sets :
If X = {3, 6, 9, 12, 15};
& Y = {4, 8, 12, 16};
A ∪ B = {3, 4, 6, 8, 9, 12, 15, 16};
X
28
2
7
Y
8 4
5 3
13 9
X
3 6 12
9
15
Y
8
4
16

9. Proving the Distributive Law of Sets
The Distributive Law of Sets is :
i. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C )
ii. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C )
Let us prove it by Venn diagrams.
Let's take three sets : A, B and C
A B
C

10. (i)
Step : 1 (LHS)
(B ∪ C)
A
B C
ξ

11. Step : 2 (LHS)
A ∩ (B ∪ C)
Fig. 1
ξ
A
B C

12. Step : 3 (RHS)
(A ∩ B)
ξ
A
B C

13. Step : 4 (RHS)
(A ∩ C )
ξ
A
B C

14. Step : 5 (RHS)
(A ∩ B) ∪ (A ∩ C )
Fig. 2
ξ
A
B C

15. (ii)
Step : 1 (LHS)
(B ∩ C)
ξ
A
B C

16. Step : 2 (LHS)
A ∪ (B ∩ C)
Fig. 3
ξ
A
B C

17. Step : 3 (RHS)
(A ∪ B)
ξ
A
B C

18. Step : 4 (RHS)
(A ∪ C )
ξ
A
B C

19. Step : 5 (RHS)
(A ∪ B) ∩ (A ∪ C )
Fig. 4
ξ
A
B C

20. .
Fig. 1 is same as Fig. 2
& Fig. 3 is same as Fig. 4
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C )
& A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C )
PROVED

21. Bibliography
• ISC Mathematics by S Chand
• Teachoo.com
• Mathisfun.com
• Basicmathematics.com
• Onlinemathlearning.com

22. THANK YOU