Matematika terapan minggu ke-3

TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

Week 3

SET THEORY

(Continued)

SET THEORYSET THEORY

OBJECTIVES:

1. Subset and superset relation

2. Cardinality & Power of Set

3. Algebra Law of Sets


3. Algebra Law of Sets

4. Inclusion

5. Cartesian Product


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?


They maen……

S⊆T means that every element of S is also an element of T.


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


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,


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}


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??














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


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 }


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 }

Diagram Vennya adalah sbb:


16

17

18

1920

MN

13

14 15

S

11

12

Diagram Vennya adalah sbb:

SET THEORYSET THEORYAlgebra Law of Sets





Set’s Inclusion and Exclusion

For A and B, Let A and B be any finite sets. Then :

A ∪ B = A + B – A ∩ B


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.

Inclusion Exclusion


Set’s Inclusion and Exclusion


A ∪ B = A + B – A ∩ B



Inclusion Exclusion


Inclusion and Exclusion of Sets


A ∪ B = A + B – A ∩ B



Inclusion Exclusion

Inclusion-Exclusion Principle

• 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}

Contoh:

Dari 60 siswa terdapat 20 orang suka bakso, 46 orang suka siomay dan 5 orang tidak suka keduanya.a. Ada berapa orang siswa yang suka bakso dan siomay?

b. Ada berapa orang siswa yang hanya suka bakso?

c. Ada berapa orang siswa yang hanya suka siomay?

Jawab: N(S) = 60

Misalnya : A = {siswa suka bakso} n(A) = 20

B = {siswa suka siomay} n(B) = 46

(A ∩∩∩∩B)c = {tidak suka keduanya} n((A ∩∩∩∩B)c) = 5


Maka A ∩∩∩∩B = {suka keduanya}

(A ∩∩∩∩B)c = {tidak suka keduanya} n((A ∩∩∩∩B)c) = 5

n(A ∩∩∩∩B) = x

{siswa suka bakso saja} = 20 - x

{siswa suka siomay saja} = 46 - x

Perhatikan Diagram Venn berikut

xA B20 - x 46 - x

S

5

n(S) = (20 – x)+x+(46-x)+5

60 = 71 - x

X = 71 – 60 = 11a. Yang suka keduanya adalah x

= 11 orangb. Yang suka bakso saja adalah

20-x = 20-11= 9 orang

c. Yang suka siomay saja adalah

46-x = 46-11= 35 orang


Berapa banyaknya bilangan bulat antara 1

dan 100 yang habis dibagi 3 atau 5?


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 for finite A, B, |A×B|=|A||B|.

• Note that the Cartesian product is not

commutative: A×B ≠ B×A.

Education

Matematika terapan minggu ke-3