3. What is a set?
• A well- defined, unordered collection or
group of objects of any kind
– The set of students in the Math 11 section O
class is an example of a set
– The set of subjects in the BSB program
Question: Is the group of smart students in
the CMSC 56 class a set?
14 June 2011 3Math 11 College Algebra
4. Famous Set Contributors
Georg Ferdinand Ludwig
Philipp Cantor
(1845 – 1918)
German mathematician
who made the first
formal study on sets;
published main paper
on sets in 1874
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5. Georg Cantor
• set of integers had an equal number of
members as the set of even numbers,
squares, cubes, and roots to equations
• number of points in a line segment is equal
to the number of points in an infinite line, a
plane and all mathematical space
14 June 2011 5Math 11 College Algebra
6. Famous Set Contributors
Augustus De Morgan
(1806-1871)
Responsible for the
idea of a universal
set
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7. Augustus De Morgan
• first person to define and name
"mathematical induction"
• developed De Morgan's rule to determine
the convergence of a mathematical series
• Formal Logic, his most important work
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8. Famous Set Contributors
John Venn
(1834 – 1923)
Considered the universal
set as the field of vision
Responsible for the
pictorial representation of
sets, the “Venn Diagram”
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9. Two Kinds of Sets According
to Size
• Finite sets – sets whose elements can be
enumerated or exhausted
Example: The set of BS Architecture
students in UP Mindanao
• Infinite sets- sets whose elements cannot
be itemized
Example: The set of whole numbers in the
real number system
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10. Symbols for Sets
1. Naming a Set: Capital letters are used to
name sets
2. Describing a Set: Two methods can be used
to describe a set
a. Roster Method – listing the elements of the
set
b. Rule Method – defining the set by the
common characteristic/s of the elements of
the set
14 June 2011 10Math 11 College Algebra
11. Symbols for Sets
• Membership in a Set – The symbol ∈ means
that the element is a member of the set, while
∉means the element is not a member of the set
Example: Let A = {marker, ballpen, pencil}
We can say that marker ∈A, while paper ∉A
14 June 2011 11Math 11 College Algebra
12. Symbols for Sets
• Empty Set or Null Set – refers to sets with
no elements, denoted by ∅ or { }
• Universal Set – the set of all objects under
consideration. This is usually denoted by U.
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13. Set Operations
• When we have two sets A and B, the
operations enumerated below can be applied:
1. Union – the set of all elements in the
universal set U, which belong in A or B
(A∪B)
Example: A = {1,2,3,4,5} , B = {2,4,6,8}
A∪B = {1,2,3,4,5,6,8}
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14. Set Operations
2. Intersection – the set of all elements in the
universal set U, which belong in both A and
B. (A∩B)
Example: A∩B = {2,4}
3. Difference – the set of all elements in U
which belong in A, but not in B (A – B)
Example: A – B = {1,3,5}
B – A = {6,8}
14 June 2011 14Math 11 College Algebra
15. Set Operations
4. Complement – The set of all elements in
the universal set U which are not in a
certain set. (A’)
Example: U = {1,2,3,4,5,6,7,8,9,10}
A = {1,2,3,4,5}
A’ = {6,7,8,9,10}
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16. Examples:
• Given: U = {1,2,3,4,5,a,b,c,d,6,7,e}
A = {7,3,a,b}
B = {1,2,3,6,7,c,d}
Find:
1. A∪B 3. A – B 5. A’
2. A∩B 4. B – A 6. B’
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17. Venn Diagram
• A picture representation of set relationships,
developed by John Venn (1834 – 1923).
• Properties:
– Elements of the universal set U are
represented by points in a rectangular region
– Members of sets in U are represented by
points within closed regions.
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19. U U
A BAB
A – B A’
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20. Set Relations
• Let A and B be sets in some universal set
U. Then A and B can be related according
to the following:
1. A is a subset of B: (“A ⊆ B”) - every
element of A is in B
- subsequently, the null set is a member of
every set.
14 June 2011 20Math 11 College Algebra
21. Set Relations
2. A equals B: (“A = B”) -every element of A
is also the element of B; A and B have
“identical elements”
3. A is a proper subset of B:(“A ⊂ B”)
- A is a subset of B but A ≠ B
4. A and B are disjoint: (“A ∩ B = ∅”)
-A and B have no common elements.
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22. Set Relations
5. A and B are complements: (A ∪B = U,
A ∩ B = ∅) A' = B and B' = A
6. A and B are equivalent: ( “ A~B ”) - A has
the same cardinality as B
Cardinality, |A| – number of elements in a set
Note: If A and B are disjoint: |A∪B| = |A| + |B|
If not, |A∪B| = |A| + |B| - |A ∩ B|
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23. In the Language of Venn…
U U
A B
B A
A = B A ⊆ B
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24. U
U
A ∩ B = ∅
A ~ B
A
B
A 1
2 3
B
a
b c
14 June 2011 24Math 11 College Algebra
25. Exercises
1. Fill in the blanks with the correct term:
a. A ream of______
b. A school of _____
c. A pack of ______
1. Indicate the sets in another form:
a. A = {x/x is a course in UP Mindanao}
b. B = {Davao, Digos, Tagum, Panabo}
c. C = {red, blue, yellow}
14 June 2011 25Math 11 College Algebra
26. Exercises
3. Indicate the elements of each set.
a. E = {v/v Є N, 0 < v < 3}
b. G = {a/a is a color on the Philippine
Flag}
c. H = {z/z is a Math subject of a
Comm. Arts student}
14 June 2011 26Math 11 College Algebra
27. 4. From the data compiled by the University
Registrar, it has been learned that 200 students
have enrolled in Math, 180 in History, while
170 have enrolled in English. Forty- five have
enrolled in both History and English, 50 in Math
and English and 40 in both Math and History.
Fifteen students, meanwhile, enrolled in all
three subjects. Out of all 500 students, how
many enrolled...
a.) in Math only?
b.) in both Math and English but not History?
c.) in at least one of these courses?
d.) in none of these courses?
14 June 2011 27Math 11 College Algebra
Editor's Notes
Well-defined = i.e., for any given object, it must be unambiguous whether or not the object is an element of the set
For example, if a set contains all the chairs in a designated room, then any chair can be determined either to be in or not in the set.
There must be a universe
(A union B)’ = A’ intersection B’
On the Diagrammatic and Mechanical Representation of Prepositions and Reasonings – a paper introducing diagrams (now known as venn diagrams)
paper
Fishes
Cigarettes
A={BSAM,BSBIO,BSCS,BSFT,BSARCHI,BSCOMMARTS,BAENG,BAANTHRO,BSABE,MASTER MGT,DIPLOMA URP,MASTER URP}
B=
C=
