sets Mathematics

2. A Set is any well defined collection of “objects.”
The elements of a set are the objects in a set.
The symbol of Set is “ { } ” and

3. – Distribute Sets
by listing each element
– Condition sets
by defining the rules for membership

4. We will take A , B , C , D , …, Z be the
name of sets and “{ }” is symbol of sets.
And use comma ( , ) between each elements in the set.
Example Set of day in a week.
Then we get
A = {Monday, Tuesday , Wednesday,Thursday,Friday,Saturday,Sunday}

5. We use a, b, c, d, …, z be the variable that is
instead all elements in set and following with “ | ”
to tell condition about the set.
Example Set of day in a week.
Then we get
A = { x | x is day in a week}

6. 2) set of digit
Then we get
B = {0 , 1 , 2 , 3 , 4, 5, 6, 7, 8, 9 }
B = { x | x is digit}

7. 3) set of integer
Then we get
C = {…,-3,-2,-1,0,1,2,3,… }
C = { x | x is integer}

8. 4) set of numbers that between -9 to 20
Then we get
D = {-8,-7,-6,…,18,19 }
D = { x | x is numbers that between -9 to 20}

9. 5) set of integer that is divided by 3
Then we get
G = {…,-9,-6,-3,0,3,6,9,12,… }
G = { x | x is divied by 3}
5) set of integer that is divided by 3
Then we get G = {…,-9,-6,-3,0,3,6,9,12,…}
G = { x / x is divied by 3}

10. an element is a member of a set
notation:  means “is an element of”
 means “is not an element of”

11. A = {1, 2, 3, 4}
1  A 6  A
2  A z  A
B = {x | x is an even number  10}
2  B 9  B
4  B z  B

12. C = {{0,1}, 2, {3}, 4}
1  A {0,1} A
{3}  A 3  A
B = {,2}
2  B {}  B
  B {2}  B

13. Z or I = the set of integers = {…,-3,-2,-1,0,1,2,3,…}
N = the set of nonnegative integers or natural numbers = {1,2,3,4,…}
Z+ = the set of positive integers = {1,2,3,4,…}
Z- = the set of positive integers = {-1,-2,-3,-4,…}
Q = the set of rational numbers = {a/b| a,b is integer, b not zero}
Q+ = the set of positive rational numbers
Q* = the set of nonzero rational numbers
R = the set of real numbers
R+= the set of positive real numbers
R*= the set of nonzero real numbers
C = the set of complex numbers

14. Any set that contains no elements is called the empty set
the empty set is a subset of every set including itself
notation: { } or 
Examples ~ both A and B are empty
A = {x | x is a Chevrolet Mustang}
B = {x | x is a positive number  0}

15. Cardinality refers to the number of elements in a set
A finite set has a countable number of elements
An infinite set has at least as many elements as the set
of natural numbers
notation: n(A) represents the cardinality of Set A

16. Finite sets
A = {x | x is a lower case letter} n(A) = 26
B = {2, 3, 4, 5, 6, 7} n(B) = 6
C = {x | x is an even number  10} n(C)= 4
D = {x | x is an even number  10} n(D)= 5

17. Infinite sets
A = {1, 2, 3, …} n(A) = 
B = {x | x is a point on a line} n(B) = 
C = {x| x is a point in a plane} n(C) = 

18. The universal set is the set of all things pertinent
to a given discussion and is designated by the symbol U
Example:
A = {x/x2 = 4}
1. U = set of Real numbers
then A = { -2 , 2 }
2. U = set of Natural numbers
then A = { 2 }

19. Two sets are equal if and only if they contain
precisely the same elements.
• The order in which the elements are listed is
unimportant.
• Elements may be repeated in set definitions without
increasing the size of the sets.

20. 1) A = {1, 2, 3, 4} B = {1, 4, 2, 3}
then A = B
2) A = {1, 2, 2, 3, 4, 1, 2} B = {1, 2, 3, 4}
then A = B
3) A = {…,-4,-2,0,2, 4, 6, …}
B = {x/x is even numbers }
then A = B

21. 4) A = {0,1, 2, 3} B = {1, 2, 3}
then A ≠ B
5) A = {x/x is positive numbers} B = {0,1,2,3,…}
then A ≠ B

22. a subset exists when a set’s members are also
contained in another set
notation:
 means “is a subset of”
 means “is a proper subset of”
 means “is not a subset of”

23. A = {x | x is a positive integer  8}
set A contains: 1, 2, 3, 4, 5, 6, 7, 8
B = {x | x is a positive even integer  10}
set B contains: 2, 4, 6, 8
C = {2, 4, 6, 8, 10}
set C contains: 2, 4, 6, 8, 10
Subset Relationships
A  A A  B A  C
B  A B  B B  C
C  A C  B C  C

24. How to find subsets
Example 1. A = {7}
Then subsets be {7} , 
2. B = {2, 4}
Then subsets be {2} ,{4},{2,4}, 

25. How to find subsets
Example 3. C = {{0,1},2}
Then subsets be {{0,1}},{2},{{0,1},2} , 
4. D = {{1}, 5}
Then subsets be {{1}} , 5, {{1},5} , 

26. How to find subsets
Example 5. D = {1,5,7}
Then subsets be {1},{5},{7},{1,5},{1,7},{5,7},{1,5,7} , 
6. F = {{2},6, 8}
Then subsets be {{2}} ,{6},{8},{{2},6},{{2},8},{6,8}, {{2},6, 8},

27. How to find subsets
give “n” is the cardinality of the set
then we get subsets of set is 2n

28. The power set is the set of all subsets that
can be created from a given set
The cardinality of the power set is 2 to the
power of the given set’s cardinality
notation: P (set name)

29. Example:
A = {a, b, c} where n(A) = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, }
and n(P (A)) = 8
In general, if n(A) = n, then n(P (A) ) = 2n

30. Example:
B = {{1}, } where n(A) = 2
P (A) = {{{1}}, {}, {{1}, }, }
and n(P (A)) = 4
In general, if n(A) = n, then n(P (A) ) = 2n

31. • Venn diagrams show relationships between sets and their elements
Universal Set
Sets A & B

32. ©1999 Indiana University Trustees
Set Definition Elements
A = {x | x  Z+ and x  8} 1 2 3 4 5 6 7 8
B = {x | x  Z+; x is even and  10} 2 4 6 8 10
A  B
B  A

33. ©1999 Indiana University Trustees
Set Definition Elements
A = {x | x  Z+ and x  9} 1 2 3 4 5 6 7 8 9
B = {x | x  Z+ ; x is even and  8} 2 4 6 8
A  B
B  A
A  B

34. ©1999 Indiana University Trustees
Set Definition Elements
A = {x | x  Z+ ; x is even and  10} 2 4 6 8 10
B = x  Z+ ; x is odd and x  10 } 1 3 5 7 9
A  B
B  A

35. Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
A = {1, 2, 6, 7}

36. ©1999 Indiana University Trustees
Venn Diagram Example 5
Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
B = {2, 3, 4, 7}

37. ©1999 Indiana University Trustees
Venn Diagram Example 6
Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}
B = {2, 3, 4, 7}
C = {4, 5, 6, 7}
C = {4, 5, 6, 7}

39. A survey of people who attended a financial amount of
100 people. Found a common interest to share with 45 people
buying lottery savings of 25 people and share their interests
into shares and buy lottery savings of 15 people. How many
people are not interested to share and not by lottery saving ?

40. A survey of people who attended a financial amount of
100 people. Found a common interest to share with 45 people
buying lottery savings of 25 people and share their interests
into shares and buy lottery savings of 15 people. How many
people are not interested to share and not by lottery saving ?

41. A survey of students who will educated amount of 1,000
people. Found some interest to study in the higher with 370
people. Some want to work with 550 people and some want
to study in the higher or work with 850 .How many students
who interested to study higher and work too ?

42. A survey of students who will educated amount of 1,000
people. Found some interest to study in the higher with 370
people. Some want to work with 550 people and some want
to study in the higher or work with 850 .How many students
who interested to study higher and work too ?

44. A survey of people who like travel in Phetchabun amount 1,000
people that found some like to travel by bus with 320 people.Some like to
travel by car with 240 and some like to travel by motorcycle with 560.
Some like to travel by bus and car with 100 people ,some like to travel by
bus and motorcycle with 50 people and some like to travel by car and
motorcycle with 40 people and some like to travel both with 30 people.
How many people who like to travel at least one way ?

45. A survey of people who like travel in Phetchabun amount 1,000
people that found some like to travel by bus with 320 people.Some like to
travel by car with 240 and some like to travel by motorcycle with 560.
Some like to travel by bus and car with 100 people ,some like to travel by
bus and motorcycle with 50 people and some like to travel by car and
motorcycle with 40 people and some like to travel both with 30 people.
How many people who like to travel at least one way ?