Set Operations Topic Including Union , Intersection, Disjoint etc De Morgans Prove With Important Examples For The Students Of B.Sc and F.Sc

1. 1
TheStuffPoint.ComTheStuffPoint.Com
Set OperatiOnS
Union Intersection
Disjoint sets
V.Imp De Morgan Laws
BY Abu Bakar SoomroBY Abu Bakar Soomro

2. 2
Set OperatiOnS
A B∪ A B∩ A B−
roc
A A′

3. 3
Set operations: Union
A
B
A B∪
U

4. 4
Set operations: Union
• Formal definition for the union of two sets:
• Further examples
{ }A B x x A x B∪ = ∈ ∈∨
{ }orA B x x A x B∪ = ∈ ∈
3 5 7 3 5 7{2, , , ,11,13} {1, , , ,9} {1,2, , , ,9,113 5 7 ,13}∪ =
{2,3,5,7,11,13} {2,3,5,7,11,13}∅ =∪

5. 5
Set operations: Union
• Properties of the union operation
 Identity law
 Domination law
 Idempotent law
 Commutative law
 Associative law
A A∪ ∅ =
Empty set=∅ Universal setU =
A U U∪ =
A A A∪ =
A B B A∪ = ∪
( ) ( )A B C A B C∪ ∪ = ∪ ∪

6. 6
Set operations: Intersection
A
B
A B∩
U

7. 7
Set operations: Intersection
• Formal definition for the intersection of two
sets:
• Further examples
{ }A B x x A x B∩ = ∈ ∈∧
{2, , , ,11,13} {1, , , ,9}3 5 7 3 5 7 7{ }3,5,=∩
{2,3,5,7,11,13}∩ ∅ = ∅
{ }andA B x x A x B∩ = ∈ ∈

8. 8
Set operations: Intersection 4
• Properties of the intersection operation
 Identity law
 Domination law
 Idempotent law
 Commutative law
 Associative law
A U A∩ =
A∩ ∅ = ∅
A A A∩ =
A B B A∩ = ∩
( ) ( )A B C A B C∩ ∩ = ∩ ∩

9. 9
Disjoint sets
A B
U

10. 10
Disjoint sets
• Formal definition for disjoint sets: two sets
are disjoint if their intersection is the empty
set.
• i.e.
• Further examples
 {1, 2, 3} and {3, 4, 5} are not disjoint
 {1, 2} and ∅ are disjoint
• Their intersection is the empty set
 ∅ and ∅ are disjoint!
• Their intersection is the empty set
A B∩ = ∅

11. 11
A
B
A B−
A B′= ∩
U
Set operations: Difference

12. 12
• Formal definition for the difference of two
sets:
• Further examples
Set operations: Difference
{ }A B x x A x B− = ∈ ∉∧
{ , , , , , } {1,3 5 7 3 5 7, , ,9}2 11 13 2 11 3{ , , }1− =
{ }andA B x x A x B− = ∈ ∉
c
A B A B− = ∩
c
A A A U A′= = = −

13. 13
• Formal definition for the symmetric difference of
two sets:
Further examples
Set operations: Symmetric
Difference
{ }A B x x A B x A B⊕ = ∈ ∪ ∉ ∩∧
( ) ( )A B A B A B⊕ = ∪ − ∩
( ) ( )A B A B B A⊕ = − ∪ −
{2, , , ,11,13} { , , , , } {2,13 5 7 3 5 1,13,7 9}1 9 1,⊕ =

14. 14
A
B
A B−
U
B A−

15. 15
Complement sets
A
B
c
B U
c
A U A= −

16. 16
Complement sets
• Formal definition for the complement of a
set c
A A U A′= = −
1,2,3,... 0{ },1U =
,3,2 }7{ 5,A =
,4, ,6,8 }0{2 5 ,1B =
1, , ,4, ,6, ,8,9,10{ , , ,2 3 5 7 2 3 5} }7{c
A U A −= − =
1,4,6,8,9 1{ }, 0c
A =

17. 17
{ ,7}3A B− =
4,6,8{ , 0}1B A− =
1,2,3,... 0{ },1U =
,3,2 }7{ 5,A =
,4, ,6,8 }0{2 5 ,1B =
2 5,3,4, ,6,7,8 1{ }, 0A B∪ =
{ 5, }2A B∩ =
Exp.:

18. 18
De Morgan Laws
• For any
we have
( ) ,c c c
A B A B= ∩∪
,A B U⊆ ≠ ∅
( )c c c
A B A B= ∪∩

19. Q:
19
1,2,3,... 0{ },1U =
,3,2 }7{ 5,A =
,4, ,6,8 }0{2 5 ,1B =
2 5,3,4, ,6,7,8 1{ }, 0A B∪ =
{ 5, }2A B∩ =
1,4,6,8,9,10{ }c
A U A= − =
1,3,7,9{ }c
B U B= − =
Verify De Morgan’s Laws

20. 20
( ) ,9( ) 1{ }c
U A BA B ∪= − =∪
1,3,4,6,7,8,9,1{( ( ) 0})c
A U A BB ∩∩ = − =
1,3,4,6,7,8,9,10{ }c c
A B∪ =
{1 },9c c
A B =∩
( )c c c
A B A B=∪ ∩
( )c c c
A B A B=∩ ∪

21. Order of
21
A B∪

22. 22
A B A B A B= + −∪ ∩
Exp.: ,3,2 }7{ 5,A =
,4, ,6,8 }0{2 5 ,1B =
2 5,3,4, ,6,7,8 1{ }, 0A B∪ =
U
B
A B∩
A B∪
A
2
5
3
7
4
6
8
10
8 4 6 2= + −

23. Q:
Each student in a class of 45 students can
speak either Urdu or English. If 25 of the
students can speak Urdu and 15 can speak
both, find, analytically, the number of those
who can speak
(i) English, (ii) English only, (iii) Urdu only?
23

24. 24
U E
15
25

25. 25
U E
15 20
25
10
U E∩
U E- E U-

26. 26
U E
35
15 20
25
10
U E∩
U E- E U-

27. 27
( ) 25, ( ) 15, ( ) 45.n U n U E n U E= ∩ = ∪ =
( ) ( ) ( ) ( )n U E n U n E n U E∪ = + ∩-
( ) ( ) ( ) ( ) 45 25 15 35n E n U E n U n U E= ∪ + ∩ = + =- -
( ) ( ) 35 15 20,n E n U E∩ = − =-
( ) ( ) 20 15 5.n U n U E∩ = − =-