MAthematics 522331894-G7-MATH-Sets-WEEK-1.ppt

Sets

1. SETS
Designed By;
SALGIE P. SERNAL

2. Which does not belong to the group?
Exploration

3. What is a Set?
A set is a well-defined collection of
elements, class things or objects.
The objects in a set are called the
elements or members of the set.
Capital letters A,B,C,… usually
denote sets.
Lowercase letters a,b,c,… denote
the elements of a set.

4. Examples
Set of Dishes
Set of Dictionaries.
Set of Points
Set of Integers

5. Which of the Following is a
“Well-Defined” Set?
Set of all Large Numbers?
Set of ALL multiples of 2?
Set of Good Politicians?
Set of Mathematics Notebooks?
Set of Honest Students?

6. 3 Ways to write a SET
Roster Notation/Listing Method
-describing set by listing each element of
the set inside the symbol { }. Infinite
numbers can be written in ellipsis (…) .
A= { 1,2,3,4,5 }
B = { 1,2,3,4,5, … }
C= { i,l,o,v,e,y,o,u }
D = { Jesus }

7. Practice
List all elements of elementary
grade level :
Let A be set of elementary
student
A = { 1,2,3,4,5,6 }

8. 3 Ways to write a SET
Set Builder Notation
-describing set by listing the rules that
determine whether an object is an element of
the set.
A={x x is a letter in the word Mathematics}
│
B= {x x is a positive multiple of 5}
│
C ={ x x is a country in the world }
│

9. Practice
List the elements using Set Builder
Notation/write a rule;
A = { Monday,Tuesday,…,Sunday}

10. 3 Ways to write a SET
Verbal Description Method
-describing set by words.
Set A is the set of letters in the word
Christian.
Set B is the set of positive multiples of
5
Set C is the set of mountains in the
world.

11. Practice
Write a verbal description of the
set;
A= { 2,4,6,8,…}

12. More Practices
Write a verbal description for the set.
E = {a,b,c,…,z}
Set A is all the letter of the Alphabet

13. More Practice
List all the elements using Roster
Method
M = {x | x>6, x is an odd integer }
M = { 7,9,11,… }

14. More Practice
Write a rule
M = {a,e,i,o,u }
M = { x | x is all vowels }

15. The Cardinality of a Set/Counting
Set
Notation: n(A)
For finite sets A, n(A) is the number of
elements of A.
For infinite sets A, write n(A)=∞.
A= { 1,4,3 } n(A)=3
B={1,3,6,7,9} n(A)=5

16. Equal & Equivalent Set
Equivalent Set
-sets that contain the same number of
elements.
Given A= { 1,2,3,4}
B= { L,O,V,E }
Therefore ; A ≈ B

17. Equal & Equivalent Set
Equal Set
-sets that contain exactly the same
elements.
Given A= { 1,2,3,4}
B= { 1,2,3,4 }
Therefore ; A = B

18. Seatwork
List the set of all names of the
apostles in the Bible using Roster
method, Set Builder Notation and
Verbal Description Method.

19. Assignment 1&2
Answer all ODD numbers in Practice
Application I,II,III,IV,V. p.14.

21. Sets
Universal Sets, Subsets, Null Sets

22. The Empty Set
The set with no elements.
Also called the null set.

Denoted by the symbol can be
written in { } 

23. Subsets
A set of which all the elements are
contained in another set(Superset).
Notation:
A = { 1,2,3 } B= { 0,1,2,3,4,5}
A is a subset of B since every element of
A is an element of B.
A= {1,2,3 } B = { 1,2,3}
BA
BA AB AA

24. Proper Subsets
A set of which all the elements are
contained in another set(Superset).
Notation:
A = { 1,2,3 } B= { 1,2,3,4}
A is a proper subset of B since there are at
least one of elements of B does not contain
in A.
A= {1,2,3 } B = { 1,2,3}
BA
BA BA

25. Exercises
Given 
A
 
= {5, 2, 4} and 
B 
= {1, 2,
3, 4, 5}, what is the relationship
between these sets?

26. Exercises
Given 
X
 
= {a, r, e} and 
Y 
= {r, e,
a, d}, what is the relationship
between these sets?

27. Exercises
Given 
X
 
= {1 , 2 , 4} and 
Y 
=
{5,10,15,20}, what is the
relationship between these sets?

28. Exercises
Given 
X
 
= {vowels} and 
Y 
=
{alphabet}, what is the relationship
between these sets?

29. Seatwork
Let A = {2, 3, 4, 5, 6, 7} B = {2, 4, 7, 8) C = {2,
4}. Fill in the blanks by or to make the
⊂ ⊄
resulting statements true.
(a) B __ A
(b) C __ A
(c) B __ C
(d) __ B
∅
(e) C __ C
(f) C __ B

30. Finite and Infinite Sets
A finite set is one which can be
counted.
Example: The set of two-digit
positive integers has 90 elements.
An infinite set is one which cannot
be counted.
Example: The set of integer
multiples of the number 5.

31. Specifying a Set
List the elements explicitly,
List the elements implicitly,
Use set builder notation,
  , ,C a o i
  10,15,20,25,....,95K
  / where and are integers and 0Q x x p q p q q  

32. The Universal Set
A set U that includes all of the
elements under consideration in a
particular discussion.
Depends on the context.
Examples: The set of Latin letters,
the set of natural numbers, the set
of points on a line.

33. Examples
Suppose A={ 1,3,5} & B= {1,2,6}
what is set U?

34. Examples
Suppose C={ 5,6,8} &
D= {9,10,11} what is set U?

35. Examples
Suppose A={ a,b,c} &
B= {a,x,z} what is set U?

36. Examples
Suppose A={ bygrace,alex,matt} &
B= {rex,shaun,alex,nica}
what is set U?

37. The Membership Relation
Let A be a set and let x be some
object.
Notation:
Meaning: x is a member of A, or x is
an element of A, or x belongs to A.
Negated by writing
Example: . , .
Ax
Ax
  , , , ,V a ei ou Ve Vb

38. Examples
Suppose A= { 5,9,25,42,15},determine
the relationship of
5 ___ A
a ____ A
x____ A
42____ A
15____ A

39. Equality of Sets
Two sets A and B are equal, denoted
A=B, if they have the same elements.
Otherwise, A≠B.
Example: The set A of odd positive
integers is not equal to the set B of prime
numbers.
Example: The set of odd integers between
4 and 8 is equal to the set of prime
numbers between 4 and 8.

40. Unions
The union of two sets A and B is
The word “or” is inclusive.
  or A B x x A x B   

41. Examples of Union

42. Seatwork # 2

43. Intersections
The intersection of A and B is
Example: Let A be the set of even
positive integers and B the set of prime
positive integers. Then
Definition: A and B are disjoint if
  and A B x x A x B   
}2{BA
ØBA

44. Examples

45. Examples

46. Examples

47. Complements
oIf A is a subset of the universal set U,
then the complement of A is the set
oNote: ;
c
AA
 

c
A x U x A  
UAA
c


48. Examples
Given U={1,2,3,4,5}, A={1,3,5}
B={1,5} C={ }, Find
A’
B’
C’

49. Venn Diagrams

A
Set A represented as a disk inside a
rectangular region representing U.
U

50. Possible Venn Diagrams
for Two Sets

U
A B

U
A B
U
A B

51. The Union of Two Sets

U
A B

52. The Intersection of Two Sets

U
A B

53. Sets Formed by Two Sets
o



R
1

R
3
U
A B
R
2
R
4
c
BAR 
1
BAR 
2
BAR
c

3
cc
BAR 
4

54. Examples
Let U be a universal set consisting of all
the natural numbers until 20 and set A and
B be a subset of U defined as
A={2,5,9,15,19} and B = {8, 9, 10, 13, 15,
17}. Find A B.
∪
Solution: 
Given
U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,
18,19,20}
A={2,5,9,15,19}
B={8,9,10,13,15,17}
A B={2,5,8,9,10,13,15,17,19}
∪

55. A={2,5,9,15,19}
B={8,9,10,13,15,17}
A B={2,5,8,9,10,13,15,17,19}
∪

56. Examples
A= {a,b,c} B={c,d,e}, Find A B
∪

57. Examples
A= {jollibee,mcdo,kfc} B={kfc,wendys,goldilocks} ,Find A
B
∪

58. Seatwork
Find the union and represent the following
into Venn Diagrams.
U = {1,2,3,4,5,6,7}
A= {2,4,6,7}
B= {1,2,4,5,7}