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Set Theory
Null, Singleton, and Universal Set
What is a Set?
Writing Sets
Relationship among Sets
Notes on the Concept of Sets
Operation on Sets
G0001
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What is a Set ?Set
*Property of STI G0001
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What is a Set ?Set is a collection of
distinct objects.
*Property of STI G0001
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is a collection of
distinct objects.Set September, October,
November, December
*Property of STI G0001
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Element is a particular
member of a set.Cardinality is the number of
elements in a set.*Property of STI G0001
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Writing Sets
*Property of STI G0001
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Two ways of writing sets:
1. Roster Method
2. Set Builder Notation
*Property of STI G0001
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Example
{September, October, November, December}
{Monday, Tuesday, Wednesday, Thursday,
Friday, Saturday, Sunday}
{Pres. Ramos, Pres. Estrada, Pres. Arroyo
Pres. Aquino}
{1, 2, 3, 4, 5, . . . }
{. . . −6, −3, 0, 3, 6, . . . }
*Property of STI G0001
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Set Builder Notation
*Property of STI G0001
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Example
{September, October, November, December}
{x | x is a month ending in “ber”}
ROSTER METHOD:
SET BUILDER NOTATION:
the set of all x such that x
is a month ending in “ber”such that
*Property of STI G0001
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Example
{1, 2, 3, 4, 5, . . . }
{x | x is a positive integer}
ROSTER METHOD:
SET BUILDER NOTATION:
{x | x is an integer and x > 0}
*Property of STI G0001
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Example
{2, 4, 6, 8, 10, . . . }
{x | x is a positive even number}
ROSTER METHOD:
SET BUILDER NOTATION:
{x | x = 2k , where k is a positive integer}
*Property of STI G0001
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Membership
*Property of STI G0001
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A
a ∈ A
a
a is an element of A*Property of STI G0001
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A
m ∉ A
m
m is not an element of A*Property of STI G0001
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Example
{September, October, November, December}B =
September
August
April
November
∈ B
∈ B
∉ B
∉ B
*Property of STI G0001
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CheckPoint
C = {1, 2, 3, 4, 5}
SET BUILDER NOTATION:
C = {x | x is an integer and 1 ≤ x ≤ 5}
136 ∈ C ∈ C ∉ C 7∉ C *Property of STI G0001
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CheckPoint
ROSTER METHOD:
D = {. . .−
10,−
5, 0, 5, 10, . . .}
2510012 ∈ D ∈ D ∉ D 16 ∉ D
*Property of STI G0001
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Exercise
1. A = {. . . −3, −2, −1, 0, 1, 2, 3, . . . }
2. B = {x | x is an even number and 3 < x < 9}
5. E = {−2, −1, 0, 1, 2}
4. D = {x | x is an odd number}
3. C = {2, 3, 5, 7, 11, 13, . . .}
*Property of STI G0001
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Notes on the
Concept of Sets
*Property of STI G0001
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Sets are invariant under the
following:
1. ordering of elements
2. repetition of elements
3. properties defining the set
*Property of STI G0001
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ordering of elements{1, 2, 3} is the same as {2, 1, 3}
{a , b , c } is the same as {c , b , a }
*Property of STI G0001
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{1, 2, 3} is the same as {2, 1, 3}
{a , b , c } is the same as {c , b , a }
ordering of elements
{1, 1, 2} is the same as {1, 2}
{b , b , b } is the same as {b }
*Property of STI G0001
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repetition of elements
{1, 1, 2} is the same as {1, 2}
{b , b , b } is the same as {b }
{x | x is a positive even number}
is the same as
{x | x = 2k, where k is a positive integer}
*Property of STI G0001
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Null, Singleton, and
Universal Set
*Property of STI G0001
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Null Set is a set with
no element.
Notation: or { }
*Property of STI G0001
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Singleton
Set
is a set with
one element.
Example:
{a} {1}
{ x | x is an integer and 1 < x < 3}
*Property of STI G0001
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Universal
Set
is a set containing
all elements under
consideration
Notation: U
*Property of STI G0001
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Relationship
among Sets
*Property of STI G0001
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U
A BVenn Diagram=
*Property of STI G0001
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U
A Bdisjoint
A and B are said to be disjoint if A and B has no
common elements.
Let A and B be sets.
efinition
A = {1, 2, 3, 4, 5}
U = {x | x is a real number}
Example
B = {6, 7, 8, 9, 10}
A = {a , b , c } B = {e , f , g , h , k }
A = {x | x is a letter and x is a consonant}
B = {x | x is a letter and x is a vowel}*Property of STI G0001
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U
A Bintersecting
A and B are said to be intersecting if A and B have at
least one common element.
Let A and B be sets.
efinition
A = {2, 4, 6}
U = {x | x is a real number}
Example
B = {3, 6, 9, 12, 15}
A = {a, b, c }
A = {x | x is a prime number}
B = {b , c, d, e , g }
B = {x | x is an odd number}*Property of STI G0001
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U
A = Bequal
A and B are said to be equal if A and B have the same
elements.
Let A and B be sets.
efinition
A = {a, b, c }
A = {x | x is an even number }
Example
B = {b , c, a }
B = {x | x is divisible by 2}*Property of STI G0001
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UAproper subset
proper superset
Let A and B be sets. B is a proper subset of A (or A is a proper
superset of B ) if:
efinition
B
a. all elements of B is in A
b. not all elements of A is in B
B ⊂ A or A ⊃ B
Notation
A = {a, b, c }
A = {x | x is an integer }
Example
B = {a , b, c, d, e }
B = {x | x is an even number}*Property of STI G0001
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subset / superset
B is a subset of A
B ⊂ A
A is a superset of B
B = A
Let A and B be sets.
efinition
B is a subset of A (or A is a superset of B ) if the following
condition is satisfied:
All elements of B is in A.
*Property of STI G0001
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B ⊆ Aotation
subset / superset
Let A and B be sets.
efinition
A is equal to B if the following conditions are satisfied:
a. A ⊆ B
b
. B ⊆ A
A ⊇ B or
*Property of STI G0001
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CheckPoint
C = {2, 4, 6, 8 . . .} D = {3, 6, 9, 12, 15 . . .}
INTERSECTING
*Property of STI G0001
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CheckPoint
DISJOINT
*Property of STI G0001
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Exercise
1. A = {x | x is an integer}B = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}
3. E = {2, 3, 4, 5, 6, . . .}
D = {. . . −6, −3, 0, 3, 6, . . .}2. C = {. . . −5, −2, 1, 4, 7, 10, . . .}
B ⊂ A
D = {x | x is an integer and x > 1}
4. E = {x | x is a prime number}D = {x | x is an even number}
disjoint
equal
intersecting
*Property of STI G0001
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Operation on
Sets
*Property of STI G0001
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A
C
B
O
*Property of STI G0001
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Set Operations
1. Union
2. Intersection
3. Relative Complement
(A ∪ B )
(A B )
(A B )
*Property of STI G0001
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Example:
A = {1, 2, 3} B = {3, 4}
A ∪ B = {1, 2, 3, 4}
D = {a , b, c, d } G = {c , d , e , f , g }
D ∪ G = {a , b , c , d , e , f , g }
W = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}
W ∪ Q = {x | x is an integer}
Q = {. . . −5, −3, −1, 1, 3, 5, . . .}
*Property of STI G0001
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Intersection (A B )
*Property of STI G0001
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Example:
A = {1, 2, 3} B = {3, 4}
A B = {3}
D = {a , b, c, d } G = {c , d , e , f , g }
D G = {c , d }
W = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}
W ∪
Q = { } =
Q = {. . . −5, −3, −1, 1, 3, 5, . . .}
*Property of STI G0001
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Relative Complement (A B )
*Property of STI G0001
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Example:
A = {1, 2, 3} B = {3, 4}
A B = {1, 2}
D = {a , b, c, d } G = {c , d , e , f , g }
D G = {a , b }
W = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}
W Q = W
Q = {. . . −5, −3, −1, 1, 3, 5, . . .}
*Property of STI G0001
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Complement
*Property of STI G0001
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Example:
U = {1, 2, 3, 4, 5} A = {3, 4}
A ‘ = {1, 2, 5}
D = {x | x is consonant }
D ‘ = {a , e, i, o, u }
U = {x | x is an integer}
Q ‘ = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}
Q = {. . . −5, −3, −1, 1, 3, 5, . . .}
U = {x | x is a letter in English alphabet}
*Property of STI G0001
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CheckPoint
C = {2, 4, 6, 8 . . .} D = {1, 3, 5, 7, 9 . . .}
C D = {x | x is an integer and x > 1}
C D = { }C D = C D C = D
U = {x | x is an integer}
C ‘ = {. . . −3, −2, −1, 0} D
D ‘ = {. . . −3, −2, −1, 0} C *Property of STI G0001
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Exercise
1. A = {−3, −1, 1, 3 }
B = {−1, 0, 1, 2, 3, 4 }
U = {x | x is an integer and −5 < x < 5}
2. C = {x | x is an integer and x is divisible by 5 }
D = {1, 3, 4}
3. E = {x | x is an integer and 0 < x < 5 }
F = {x | x is an integer and x = 2k + 1, where kis an integer}
*Property of STI G0001
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Exercise
1. A = {−3, −1, 1, 3 }
B = {−1, 0, 1, 2, 3, 4 }
U = {x | x is an integer and −5 < x < 5}
A B = {−3, −1, 0, 1, 2, 3, 4}
A B = {−1, 1, 3}
A B = {−3}
B A = {0, 4}
A ‘ = {−4, −2, 0, 2, 4} B ‘ = {−4, −3, −2}
2. C = {x | x is an integer and x is divisible by 5 }
D = {1, 3, 4}
3. E = {x | x is an integer and 0 < x < 5 }
F = {x | x is an integer and x = 2k + 1, where kis an integer}
*Property of STI G0001
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Exercise
U = {x | x is an integer and −5 < x < 5}
C D = {0, 1, 3, 4}
C D = { } = ∅
C D = {0} = C
D C = D
C ‘ = {−4, −3, −2, −1, 1, 2, 3, 4}
D ‘ = {−4, −3, −2, −1, 2}*Property of STI G0001
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Exercise
E F = {−
3,−
1, 1, 2, 3, 4}
E F = {1, 3}
E F = {2, 4}
F E = {−3, −1}
E ‘ = {−4, −3, −2, −1, 0}
F ‘ = {−4, −2, 0, 2, 4}
U = {x | x is an integer and −5 < x < 5}
*Property of STI G0001
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UA \ B B \ AA ∩ B
A ∪ B
A B
*Property of STI G0001
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U
A
= U \ AA‘*Property of STI G0001
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A B
C*Property of STI G0001
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CheckPoint
A = {1, 2, 3, 4}
A B
13 B = {2, 4, 6, 8, 9}2468911*Property of STI G0001
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A = {3, 6, 9 . . .}
A B
0 4B = {0, 2, 4, 6 . . .}
3 6 109 13 18
Exercise
*Property of STI G0001
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A B
C
A = {1, 2, 3, 4}
B = {2, 3, 4, 5}
C = {3, 4, 5, 6}
1 2
34
56
*Property of STI G0001
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A B
C
A = {a, b, c}
B = {c, d, e, f}
C = {b, c, e}
ab
cd
e f
*Property of STI G0001
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End
*Property of STI G0001
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