Distributive property of sets (class 11 mathematics project)

distributive law of sets with example ; ISC

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 : A ∩ (B  ∪  C) = (A ∩ B)  ∪  (A ∩ C )   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