TIF APPLIED MATH 1 (MATEMATIKA TERAPAN 1) Week 3 SET THEORY (Continued)

TIF 21101 APPLIED MATH 1 (MATEMATIKA TERAPAN 1) Week 3 SET THEORY (Continued) 2014/2015

M. Ilyas Hadikusuma, M.Eng

Matematika Terapan 1

SET THEORY OBJECTIVES: 1. Subset and superset relation 2. Cardinality & Power of Set 3. Algebra Law of Sets 4. Inclusion 5. Cartesian Product

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SET THEORY 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? 2014/2015



SET 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 2014/2015


. Matematika Terapan 1

SET 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⊄BC⊆C

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Prove them !!!



SET 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}}. 2014/2015



SET 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?? 2014/2015



SET THEORY

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SET THEORY

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SET THEORY

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SET THEORY

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SET THEORY

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SET THEORY

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SET 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 2014/2015



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

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SET 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 } Diagram Vennya adalah sbb:

S N 11 12 2014/2015

16 18 13

17 19

20

M

14 15 M. Ilyas Hadikusuma, M.Eng


SET THEORY Algebra Law of Sets

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SET THEORY

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SET THEORY Set’s Inclusion and Exclusion For A and B, Let A and B be any finite sets. Then :

A ∪ B = A + B – A ∩ B Inclusion

Exclusion

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. 2014/2015



SET THEORY Set’s Inclusion and Exclusion For A and B, Let A and B be any finite sets. Then :


Exclusion




SET THEORY Inclusion and Exclusion of Sets For A and B, Let A and B be any finite sets. Then :


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}

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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} B = {siswa suka siomay}

n(A) = 20 n(B) = 46

(A ∩B)c = {tidak suka keduanya} n((A ∩B)c) = 5 Maka A ∩B = {suka keduanya} n(A ∩B) = x n(S) = (20 – x)+x+(46-x)+5 {siswa suka bakso saja} = 20 - x 60 = 71 - x X = 71 – 60 = 11 a. Yang suka keduanya adalah x Perhatikan Diagram Venn berikut = 11 orang S b. Yang suka bakso saja adalah 20-x = 20-11= 9 orang A x B 20 - x 46 - x c. Yang suka siomay saja adalah 2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1 46-x = 46-11= 35 orang 5

{siswa suka siomay saja} = 46 - x

SET THEORY Berapa banyaknya bilangan bulat antara 1 dan 100 yang habis dibagi 3 atau 5?

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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.

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TIF APPLIED MATH 1 (MATEMATIKA TERAPAN 1) Week 3 SET THEORY (Continued)

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