Dive into the world of Venn diagrams in Discrete Mathematics with this insightful PowerPoint presentation. This presentation is designed to demystify the use of Venn diagrams, providing a clear understanding of their applications in set theory and logical reasoning.

2. Sets
CSCE 235, Fall 2010 2
Venn Diagram: Example
• A set can be represented graphically using a
Venn Diagram
U
a
x y
z
A
C
B

3. Sets
CSCE 235, Fall 2010 3
Set Complement
• Definition: The complement of a set A,
denoted A ($bar$), consists of all elements not
in A.
A= AC = {x | x  A }
U A
A

4. Sets
CSCE 235, Fall 2010 4
1/26/2024 CS 201 4
Set complement
• The venn diagram

5. Sets
CSCE 235, Fall 2010 5
Set Operators: Union
• Definition: The union of two sets A and B is
the set that contains all elements in A, B, r
both. AB = { x | (a  A)  (b  B) }
• Case III AB A and B are disjoint set
U
A B

6. Sets
CSCE 235, Fall 2010 6
1/26/2024 CS 201 6
Set union
• An example
{1, 2, 3}  {1, 2, 4, 5} = {1, 2, 3, 4, 5}
The venn diagram
1
2
3 4
5

7. Sets
CSCE 235, Fall 2010 7
Set Operators: Intersection
• Definition: The intersection of two sets A and
B is the set that contains all elements that are
element of both A and B. A  B = { x | (a  A)
 (b  B) }
Case III A  B A and B disjoint set
U
A B

8. Sets
CSCE 235, Fall 2010 8
1/26/2024 CS 201 8
Set intersection
• An example
{1, 2, 3}  {1, 2, 4, 5} = {1, 2}
The venn diagram
1
2
3 4
5

9. Sets
CSCE 235, Fall 2010 9
Disjoint Sets
• Definition: Two sets are said to be disjoint if
their intersection is the empty set: A  B = 
U
A B

10. Sets
CSCE 235, Fall 2010 10
Set Difference
• Definition: The difference of two sets A and B,
denoted AB ($setminus$) or A−B, is the set
containing those elements that are in A but
not in B
U
A B

11. Sets
CSCE 235, Fall 2010 11
• Case I A-B A s B [A-B=0]
• Case II A-B B s A
• Case III A-B A ns B or B ns A
• Case Iv A-B A and B are disjoint set

12. Sets
CSCE 235, Fall 2010 12
1/26/2024 CS 201 12
Set difference
• An example
{1, 2, 3} - {1, 2, 4, 5} = {3}
The venn diagram
1
2
3 4
5

13. Sets
CSCE 235, Fall 2010 13
1/26/2024 CS 201 13
Some questions
• Let A  B.
– What is A – B?
– What is B – A?
– If A, B  C, what can you say about A  B and C?
– If C  A, B, what can you say about C and A  B?

14. •Venn Diagram

15. Using the numbers 0, 1, 2, … , 9 illustrate
the sets: and
Solution: Use a Venn diagram

16. Using the numbers 0, 1, 2, … , 9 illustrate
the sets: and
A B
4 is in BOTH sides
4

17. Using the numbers 0, 1, 2, … , 9 illustrate
the sets: and
A B
7 and 9 are only in set A
4
7
9

18. Using the numbers 0, 1, 2, … , 9 illustrate
the sets: and
A B
1, 2, 3 and 5 are only in set B
4
7
9
1
2
3
5

19. Using the numbers 0, 1, 2, … , 9 illustrate
the sets: and
A B
0, 6 and 8 are not in A or B
4
7
9
1
2
3
5
0
6
8

20. A B
Intersection: Members of both set A and set B

21. A B
Union: Members of set A or set B or both

22. Complementary: Members not in the set
A
A’

23. Universal Set: All members
U

24. A
B
Subset: All members of set A are in set B

25. A B
U

26. A B
U

27. A B
U

28. A B
U

29. A B
U

30. A B
U

31. A B
U

32. A B
U

33. A B
U

34. A B
U

35. A B
U
What is the shaded region?

36. A B
U
What is the shaded region?

37. A B
U
What is the shaded region?

38. A B
U
What is the shaded region?
C
)
C
A
(
B 


39. A B
U
What is the shaded region?
C
)'
C
B
A
( 


40. A B
U
What is the shaded region?
C
)
B'
C
(A 


41. A B
U
1,2 4,5
3
6,7

43. A B
C
A B
C
A B
C
A B
C
A B
C

44. A B
C
A B
C
A B
C
A B
C
A B
C

45. A B
C
A B
C
A B
C
A B
C
A B
C

46. A B
C
A B
C
A B
C
A B
C
A B
C

47. A B
C
A B
C
A B
C
A B
C

48. A B
C
A B
C
A B
C
A B
C

49. U = {Natural Numbers less than 16}
Describe set A and set B
A = {Even Numbers}
B = {Prime Numbers}
A B
4
10
14
6
12
8
2
13
5
7
3
11
1
15
9
U

50. U
28
20
14
16
26
12
18
24
30
10
25
15
21
27
13
19
11
29
17
Describe Sets U, A, B and C
U = {10,11,12,13,14,........29,30}
A = {Even Numbers}
B = {Multiples of 3}
C = {Multiples of 5}
C
B
A
22
23

52. • In class of 50 students 18 takes tabla 26
takes band, & 2 take both.
• How many students in class are not
enrolled in both.
Tabla Band

53. • In school of 320 students 85 students are
in the band, 200 students are in sport
team and 60 students participate in both.
• How many students are involved in either
band or sport.
Sport
Band

54. A survey an sample a 25 new car being sold out at a local auto dealer
was conducted to see which of 3 popular option AC(A), Radio(R), and
Power window(PW) were already installed.
The survey found 15 had AC, 12 R, 11 PW.
5 had AC and PW, 9 has AC and R, 4 had R and PW.
3 had all 3 option.
Find out of car had.
1. Only one of option
2. At least one option
3. None of option
AC R
PW

55. • During a survey of Ice Cream perform by students.
• It was found that 22 like Mango, 25 like Apple, 39 like Grapes, 9 like Apple and Mangoes.
• 17 Like Mango and Grapes
• 20 like Apple and Grapes
• 6 like all
• 4 like none.
• How many students were in survey.
• How many students like exactly one flavour.
• How many students like exactly two flavours.
A M
G