Chapter-2-Symbols-and-SET-Theory.pdfffff

Mathematics

1. SYMBOLS
and
SET
Theory
1
Preparedby:ARMANDO C. MANZANO

2. Sets
Definition of aSet
Methodsofnamingaset
Properties ofSets
Operation onSets
VennDiagram
Mathematical
Language&
Symbols
21

3. 22

4. Basics of Sets
•Georg Cantor-introduced the study of sets
•Naming a Set-use capital letter of the English Alphabet
•Elements ∈-denote membership of a set
• ∉means “not an element”
•Let A be set of PSUBC student
• B is set of letters a b c ∈B but 5∉B
C is a course of PSU Bayambang

6. Describing a Set
1. Word Description—
•A be a set of PSUBC student
•B is set of letters
•C is a course of PSU Bayambang
•D BPA student of PSU Bayambang
•P is BPED student of PSUBC
•2. Listing Method or Roster Method-list all the elements of
set, separated by commas and enclose by braces
•Let R be colors of rainbow
•�={���,������,������,�����,??????��??????��,�??????����,����}
•C={BPA, BSBA, ICT, BSN, BSE, BEE, BPED, BTLED,
BECED,ABEL}

7. 3. Rule Method or Set-builder form= using the description {x |
x is ….} read as x such that x is…
•�={�|�??????����������??????����}
•�={�|�??????�������??????���??????���������}
•�={�|�??????���������������ℎ�����}
•
•Let �={�|�??????�������������������������}use Roster
method to describe
•PioloRamsay Avelino
•N={ A, P, I, O, L, M, E, N, S,V, Y, R}
•4. Modified Roster
• �={�|�??????�������������}in modified rosteras
�={2,4,6,8,…}
• �=��??????�����??????�??????����������, then
• �={11,13,15,17,19,21,…,97,99}

8. Set Operations

9. Kinds of Sets
1. Unit set-set contains exactly one element
2. Empty or Null Set-{ } or ∅Set contains no element at all, {∅}not null
3. Finite Set-set contains countable element
4. Infinite Set–set contains uncountable elements
Cardinality of sets –number of elements in a given set
Let �given setthen �(�)is cardinality of set A
5. Universal Set U-totality of all elements under consideration
Augustus De Morgan-he was the one who introduced
universal set
•Let�={�|�??????���??????���������}
• �={�|�??????�������������}
• �={�|�??????��??????�??????����??????�������??????��������??????��������}
•��=7finite
•��=∞, infinite
•��=10, finite

10. Set Relations
1.Joint Sets-sets contains at least one common element
2. Disjoint Sets-sets contains no common element at all
3. Equivalent Sets(∼)–sets contains same cardinal or
number of elements
4. Equal Sets (=)–sets contains similar or common
elements
Note: All equal sets are equivalent but not all equivalent
sets are equal

11. Subsets
•Subsets-a set is a subset of any sets when all its elements contained on the sets
•Let A, B and C is a given set and U as universal set
•⊂:proper subset –when a subset has a cardinality lesser the set given
•⊆:improper subset-a set itself and null set
•⊃ super set= the mother set
*Null set is a subset of any set
*The set itself is a subset of itself
•
•Let ??????={1,2,3,4,5,6,7,8}
• �={1,�,�,�}
• �={3,4,5,6}
• �={1,3,5,7}
• �={4,8}
•We say now that �,�,�,�⊂??????
•While null set and U itself is an improper subset (U ⊆U, ∅⊆U)

12. Power Set = set contains all possible subsets of a given sets
Let �be a given set then ℘�is power set of A
•With respect to A= {1,2,3,4}, we have subsets { 1 }, { 2}, { 3 },
{4}, {1, 2}, {1, 3}, {1, 4},{2, 3}, {2,4}, {3,4}, {1, 2,3}, {1, 2, 4}, {1,
3, 4}, {2, 3, 4} are proper subsets of A, then we also say that A
is the super set of the sets
•While ∅and {1,2,3,4}are improper subsets of A
•Using the above example, let �={1,2,3,4}then
•℘�={{ 1 }, { 2}, { 3 }, {4}, {1, 2}, {1, 3}, {1, 4},{2, 3}, {2,4},
{3,4}, {1, 2,3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1,2,3,4}, { }}
•2^4=16
•{1,2,3}=2^3=8

13. •B= {�,�,�}

14. Set Operation
Let U be universal set and �,�,�,�⊂??????
1. Union of sets (∪)-set contains either elements of two given
sets
• �∪�=��∈����∈�=��∈�∨�∈�
2. Intersection of Sets (∩)-set contains common elements of
the two given sets
• �∩�=��∈�����∈�=��∈�∧�∈�
3. Difference of Sets (–)-set contain elements not found on
other given set
• �–�=��∈�����∉�
• �–�=��∉�����∈�
4. Complement of a Set (′)-set contain elements not found on
the given set but elements of the universal set
• �′=��∉�����∈??????
• �′=��∉�����∈??????

15. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
1. Complement of : A, B, C, D, E, F�
′
={�,�,�,�}
2. Union: �∪�, �∪�, �∪�, �∪�, �∪��∪�, �∪�,
3. Intersection: �∩�, �∩�, �∩�, �∩�, �∩��∩��∩
�
4. Difference:�–�, �–�, �–�, �–�, �–�, �–�, �–�

16. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
1. Complement of : A, B, C, D, E, F�
′
={�,�,�,�}

17. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
2. Union: �∪�, �∪�, �∪�, �∪�, �∪��∪�,
�∪�,

18. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
3. Intersection: �∩�, �∩�, �∩�, �∩�, �∩��∩
��∩�

19. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
4. Difference:�–�, �–�, �–�,�–�, �–�, �–�,
�–�

20. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
1.Complement of : A, B, C, D, E, F�
′
={�,�,�,�}

21. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
2. Union: �∪�, �∪�, �∪�, �∪�, �∪��∪�,
�∪�,

22. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
3. Intersection: �∩�, �∩�, �∩�, �∩�, �∩��∩
��∩�

23. Exercise:
Let U be universal set and �,�,�,�,�,�⊂??????
Given:
??????={�,�,�,�,�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�,�}
�={�,�,�}
�={�,�}
Find
4. Difference:�–�, �–�, �–�,�–�, �–�, �–�,
�–�

25. VennDiagram
•Venn Diagram-is a set diagram introduced by John
Venn in 1880.
•He invented it to illustrate relations between a finite
collection of sets. It is also often used to illustrate
setoperations.

28. •A Venn diagram is a diagram representing mathematical or
logical sets pictorially ascircles within an enclosing
rectangle(the universal set), common elements of the sets
being represented by the areas of overlap among thecircles.

29. Venn on Operation of Sets

30. Some Illustrations on Operation on Sets

31. Solve The Problem using a Venn Diagram
A history teacher was interested to know about her class
of 42 students who keeps up with current events. She
gathered the following data:
9 students read the newspaper,
18 students listen to the radio,
30 students watch television,
3 students both read newspaper and listen to the
radio,
12, students both listen to the radio and watch
television,
6 students both read the newspaper and watch
television, and
2 students read the newspaper, listen to the radio
and watch television.
Organize the data using the Venn Diagram.

32. 9 students read the newspaper,
• 18 students listen to the radio,
• 30 students watch television,
• 3 students both read newspaper and listen to the radio,
• 12, students both listen to the radio and watch television,
• 6 students both read the newspaper and watch television, and
• 2 students read the newspaper, listen to the radio and watch
television.

33. The problem will be illustrated by working backward or
starting fro the bottom

34. Form the given Venn diagram below
•Form the given Venn diagram below
•How many students
a. use exactly one source of information
b. ate least two sources
c. either watch tvor listen to a radio
d. never use the given sources
e. either read news paper or watch tvbut not
listening to a radio
f. both watch tvand listen to a radio but not reading
newspaper

35. Form the given Venn diagram below
•Form the given Venn diagram below
•How many students
a. use exactly one source of information
b. ate least two sources
c. either watch tvor listen to a radio
d. never use the given sources
e. either read news paper or watch tvbut not
listening to a radio
f. both watch tvand listen to a radio but not reading
newspaper

36. Operations (Unary orBinary)
UnaryOperationisanoperationona
singleelement.
Example:negative of5
multiplicativeinverseof7
Binary Operation is an operation that
combinestwoelementsofasettogivea
singleelement.
e.g. multiplication 3 x 4 =12
MathematicalLanguage
&Symbols
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37. Properties of Real Numbers
(Binary Operations)
Let a, b, c and d be Real numbers
1.Closure Property
2.Commutative Property
3.Associative Property
4.Distributive Property
5.Identity
6.Inverse
Note: Refer to your book page 28-30
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38. Subsets Real numbers
1.Natural number/counting N={1,2,3,4,…
2.Whole number W={0, 1,2 ,3, 4,…}
3.Integers(Z):0, Z+=N, Z-:
Z={0, 1,-1, 2, -2,…}
4. Rational(Q): {
�
�
�,�∈�,�≠�,fractions
and decimal
5. Irrationals (Q’)
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39. Properties of Real Numbers
(Binary Operations)
Let a, b, c and d be Real numbers
1.Closure Property
a+b= real number
a(b)= real number
2. Commutative Property
a+b=b+a
axb=bxa
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40. Properties of Real Numbers
(Binary Operations)
3. Associative Property
a+(b+c)=(a+b)+c=(a+c)+b
a(b.c)=(a.b)c=(a.c)b
4.Distributive Property
a(b+c)=a(b)+a(c)
(a+b)c=a(c)+ b( c)
5. Identity a+0=a 5+___=5
a(1) =a 5x___=5
6.Inverse a+(-a) =05+___=0
a(1/a) =15x___=1
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41. Properties of Real Numbers
(Binary Operations)
3. Associative Property
a+(b+c)=(a+b)+c=(a+c)+b
2+(3+4)=(2+3)+4=(2+4)+3
a(b.c)=(a.b)c=(a.c)b
2(3.4)=(2.3)4=(2.4)3
4.Distributive Property
a(b+c)=a(b)+a(c) left hand
(a+b)c=a(c)+b(c) right hand
5. Identity a+0=a 5+___=5
a(1)=a 5x___=5
6.Inverse a+(-a) =05+___=0
a(1/a) =15x___=1
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