Upload
hardinihd
View
65
Download
6
Embed Size (px)
Citation preview
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 3
SET THEORY
(Continued)
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
OBJECTIVES:
1. Subset and superset relation
2. Cardinality, Power of Set, Venn diagram
3. Algebra Law of Sets
4. Inclusion
5. Cartesian Product
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Subset & superset relationWe use the symbols of:
⊆ � is a subset of
⊇ � is a superset of
We also use these symbols⊂ � is a proper subset of
⊃ � is a proper superset of
Why they are different?
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
They maen……
S⊆T means that every element of S is also an element of T.
S⊇T means T⊆S.
S⊂T means that S⊆T but .
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Examples:
• 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, 6, 8, 4}
• Subset Relationships
A ⊆ A A ⊄ B A ⊄ C
B ⊂ A B ⊆ B B ⊂ C
C ⊄ A C ⊄ B C ⊆ C
Prove them !!!
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Cardinality and The Power of Sets
|S|, (read “the cardinality of S”), is a measure of
how many different elements S has.
E.g., |∅|=0, |{1,2,3}| = 3, |{a,b}| = 2,
|{{1,2,3},{4,5}}| = ……
P(S); (read “the power set of a set S”) , is the set
of all subsets that can be created from given set S.
E.g. P({a,b}) = {∅, {a}, {b}, {a,b}}.
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Example:
A = {a, b, c} where |A| = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ}and |P (A)| = 8
In general, if |A| = n, then |P (A) | = 2n
How about if the set of S is not finite ? So we say S infinite.
Ex. B = {x | x is a point on a line}, can you difine them??
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORY
Venn DiagramWe use venn diagram to pictoral representation of
set
– studied and taught logic and probability theory
– articulated Boole’s algebra of logic
– devised a simple way to diagram set operations (Venn Diagrams)
John Venn 1834-1923
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Universal Set
Sets A & B
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Langkah-langkah menggambar diagram venn
1. Daftarlah setiap anggota dari masing-masing himpunan
2. Tentukan mana anggota himpunan yang dimiliki secara bersama-sama
3. Letakkan anggota himpunan yang dimiliki bersama ditengah-tengah
4. Buatlah lingkaran sebanyak himpunan yang ada yang melingkupi anggota bersama tadi
5. Lingkaran yang dibuat tadi ditandai dengan nama-nama himpunan
6. Selanjutnya lengkapilah anggota himpunan yang tertulis didalam lingkaran sesuai dengan daftar anggota himpunan itu
7. Buatlah segiempat yang memuat lingkaran-lingkaran itu, dimana segiempat ini menyatakan himpunan semestanya dan lengkapilah anggotanya apabila belum lengkap
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Diketahui : S = { x | 10 < x ≤ 20, x ∈ B }
M = { x | x > 15, x ∈ S }
N = { x | x > 12, x ∈ S }Gambarlah diagram vennya
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Jawab : S = { x | 10 < x ≤ 20, x ∈ B } = { 11,12,13,14,15,16,17,18,19,20 }
M = { x | x > 15, x ∈ S } = { 16,17,18,19,20}
N = { x | x > 12, x ∈ S } = { 13,14,15,16,17,18,19,20}
M ∩∩∩∩ N = { 16,17,18,19,20 }
16
17
18
1920
MN
13
14 15
S
11
12
Diagram Vennya adalah sbb:
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYAlgebra Law of Sets
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY
Inclusion and Exclusion of Sets
For A and B, Let A and B be any finite sets. Then :
In other words, to find the number n(A ∪ B) of elements in the union A ∪ B, we add n(A) and n(B) and then we subtract n(A ∩ B); that is, we “include” n(A) and n(B), and we “exclude” n(A ∩ B). This follows from the fact that, when we add n(A) and n(B), we have counted the elements of A ∩ B twice. This principle holds for any number of sets.
A ∪ B = A + B – A ∩ B
Inclusion Exclusion
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORY
• How many elements are in A∪B?|A∪B| = |A| + |B| − |A∩B|
• Example:
{2,3,5}∪{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORY
Exercise :
From 32 people who save paper or bottles (or both) for recycling, 30 save paper and 14 save bottles. Find the number of people who
(a) save both,
(b) save only paper, and
(c) save only bottles.
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
SET THEORYSET THEORYSET THEORYSET THEORY
Cartesian Products of Sets
• For sets A, B, their Cartesian product
A×B :≡ {(a, b) | a∈A ∧ b∈B }.
• E.g. {a,b}×{1,2} = {(a,1),(a,2),(b,1),(b,2)}
• Note that for finite A, B, |A×B|=|A||B|.
• Note that the Cartesian product is not
commutative: A×B ≠ B×A.