Matematika terapan week 3. set

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 3
SET THEORY
(Continued)

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

Subset & superset relation
We 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?

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 .

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 !!!

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}}.

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??

SET THEORYSET THEORYSET THEORYSET THEORY
Venn Diagram
We 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

Universal Set
Sets A & B

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

Diketahui : S = { x | 10 < x ≤ 20, x ∈ B }
M = { x | x > 15, x ∈ S }
N = { x | x > 12, x ∈ S }
Gambarlah diagram vennya

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
19
20
MN
13
14 15
S
11
12
Diagram Vennya adalah sbb:

Algebra Law of Sets

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

• 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}

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.

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.

Matematika terapan week 3. set

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