Teori Himpunan. Bagian III

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1 Teori Himpunan Bagian III

2 Teori Himpunan Himpunan: Kumpulan objek (konkrit atau abstrak) ) yang mempunyai syarat tertentu dan jelas, bisanya dinyatakan dengan huruf besar. a A a A a anggota dari A a bukan anggota dari A A = {a, a 2,,, a n } A memuat 2

3 Cara menyatakan himpunan a. Mendaftar b. Menyatakan sifat-sifat yang dipenuhi oleh anggota. c. Notasi pembentuk himpunan 3

4 Notasi Pembentuk Himpunan Format: {[struktur anggota]} Contoh: sedemikian hingga struktur keanggotaan] [syarat perlu untuk menjadi Q = {m/n : m,n Z, n } n Q adalah himpunan bilangan rasional Elemen-elemennya elemennya berstruktur m/n; harus memenuhi sifat setelah tanda : untuk menjadi anggota. {x R x 2 = } = {-,}{ 4

5 Contoh Himpunan: N himpunan bil. Cacah = {,,2,3,4, } P atau Z+ - himp. Bil. Bulat positif = {,2,3,4, } Z himpunan bil. bulat R himpunan bil.. real φ or {} himpunan kosong U himpunan semesta, himp.. yang memuat semua element yang dibicarakan. 5

6 A = A = {z} A = {{b, c}, {c, x, d}} Contoh Himpunan empty set/null set Note: z A, z but z {z} A = {{x, y}} Note: {x, y} A, but {x, y} {{x, y}} A = {x P(x)} set of all x such that P(x) A = {x x Nx x > 7} = {8, 9,, } set builder notation 6

7 Relasi Antar Himpunan. Himpunan yang Sama 2. Himpunan Bagian 3. Himpunan yang berpotongan 4. Himpunan Saling Lepas 5. Himpunan yang Ekuivalen 7

8 Himpunan yang Sama ( Set Equality) Himp.. A and B dikatakan sama jika keduanya memuat anggota- anggota yang tepat sama. A = B { x x A x B} atau A = B A B B A Contoh: A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = B A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} : A = {dog, cat, horse}, B = {cat, horse, dog, dog} : A B A = B 8

9 A B A B Himpunan Bagian A adalah himpunan bagian dari B jika setiap anggota A juga merupakan anggota B. A B x x (x A x B) Contoh: A = {3, 9}, B = {5, 9,, 3}, A B? A = {3, 3, 3, 9}, B = {5, 9,, 3}, A B? A = {, 2, 3}, B = {2, 3, 4}, A B? benar benar Salah 9

10 Himpunan Bagian Sifat: A = B (A B) (B A) (A B) (B C) A C (Lihat Venn Diagram) U B A C

11 Useful rules: A for any set A A A for any set A Himpunan Bagian Proper subsets (Himpunan( Bagian Sejati): A B A A is a proper subset of B B A B x x (x A x B) x x (x B x A) or A B x x (x A x B) x x (x B x A)

12 Dua himpunan A dan B dikatakan berpotongan, ditulis A)(B, jika ada anggota A yang menjadi anggota B. A)(B x x (x A x B) Himpunan A dan B dikatakan saling lepas (A//B), jika A, B, x x (x A x B) Himpunan A dan B yang Ekuivalen,, A B, A jika setiap anggota A dapat dipasangkan (dikorespondensikan) satu-satu dengan anggota B Buat Contoh Masing-masing masing!!! 2

13 Latihan. Buktikan jika M, maka M =. 2. A = {,2,3,4}; B = himpunan bilangan ganjil. Buktikan A B. 3. Buktikan A B, B C A C. 4. Buktikan K L, L M, M K K = M. 3

14 Interval Notation - Special notation for subset of R [a,b] = {x R a x b} (a,b) = {x R a < x < b} [a,b) = {x R a x < b} (a,b] = {x R a < x b} How many elements in [,]? In (,)? In {,} 4

15 Operasi Himpunan B (B complement) {x x U x B} Everything in the Universal set that is not in B A B (A union B) {x x A x B} Like inclusive or, can be in A or B or both B A B 5

16 A B (A intersect B) {x x A x B} A and B are disjoint if A B = Φ A - B (A minus B or difference) {x x A x B} A-B B = A BA A B B (symmetric difference) {x x A x B} = (A B) - (A B) We have overloaded the symbol.. Used in logic to mean exclusive or and in sets to mean symmetric difference 6

17 Contoh Let A = {n 2 n P n 4} = {,4,9,6} Let B = {n 4 n P n 4} = {,6,8,256} A B B = {,4,9,6,8,256} A B B = {,6} A-B B = {4,9} B-A A = {8, 256} A B B = {4,9,8,256} 7

18 Cardinality of Sets If a set S contains n distinct elements, n N, n we call S a finite set with cardinality n. n Examples: A = {Mercedes, BMW, Porsche}, A = 3 B = {, {2, 3}, {4, 5}, 6} B = 4 C = C = D = { x N x x 7 } D = 7 E = { x N x x 7 } E is infinite! 8

19 The Power Set P(A) power set of A A P(A) = {B B A} (contains all subsets of A) Examples: A = {x, y, z} P(A) = {,{, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}} A = P(A) = { }{ Note: A =, P(A) = 9

20 The Power Set Cardinality of power sets: P(A) = 2 A Imagine each element in A has an on/off switch Each possible switch configuration in A corresponds to one element in 2 A A x y z x y z 2 x y z 3 x y z For 3 elements in A, there are = 8 elements in P(A) 4 x y z 5 x y z 6 x y z 7 x y z 8 x y z 2

21 Cartesian Product The ordered n-tuple (a, a 2, a 3,,, a n ) is an ordered collection of objects. Two ordered n-tuples (a, a 2, a 3,,, a n ) and (b, b 2, b 3,, b n ) are equal if and only if they contain exactly the same elements in the same order,, i.e. a i = b i for i n. The Cartesian product of two sets is defined as: A B B = {(a, b) a A a b B} Example: A = {x, y}, B = {a, b, c} A B B = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)} 2

22 Cartesian Product The Cartesian product of two sets is defined as: A B B = {(a, b) a A a b B} Example: A = {good, bad}, B = {student, prof} A B B = {(good,{ student), (good, (good, prof), (bad, student), (bad, prof)} B A A = { (student, good), (prof,, good), (student, bad), (prof,, bad)} 22

23 Note that: A = A A = Cartesian Product For non-empty sets A and B: A B A A B B A A B = A B B The Cartesian product of two or more sets is defined as: A A 2 A n = {(a, a 2,,, a n ) a i A for i n} 23

24 Set Operations Union: A B A B = {x x A x x B} Example: A = {a, b}, B = {b, c, d} A B B = {a, b, c, d} Intersection: A B A B = {x x A x x B} Example: A = {a, b}, B = {b, c, d} A B B = {b} 24

25 Set Operations Two sets are called disjoint if their intersection is empty, that is, they share no elements: A B B = The difference between two sets A and B contains exactly those elements of A that are not in B: A-B B = {x x A x x B} Example: A = {a, b}, B = {b, c, d}, A-B A B = {a} 25

26 Set Operations The complement of a set A contains exactly those elements under consideration that are not in A: A c = U-AU Example: U = N,, B = {25, 25, 252, } B c = {,, 2,,, 248, 249} 26

27 Set Operations Table in Section.5 shows many useful equations. How can we prove A (BA (B C) = (A B) B) (A C)? Method I: x A (B C) x A x (B C) x A (x B x C) (x A x B) (x A x C) (distributive law for logical expressions) x (A B) x (A C) x (A B) B) (A C) 27

28 Set Operations Method II: Membership table means x x is an element of this set means x x is not an element of this set A B C B C A (B C) A B A C (A B) (A C) 28

29 Sifat Operasi Himpunan. Asosiatif: : (A B) C = A (B C) (A B) C C = A (B C) 2. Idempoten: : A A A A = A; A A A = A 3. Identitas: : A S A S = S; A A S = A A = A; A A = 4. Distributif: : A (B A C) = (A B) (A C) A (B C) = (A B) B) (A C) 5. Komplementer: : A A A = S; A A = 6. De Morgan: (A B) B) = A B A (A B) B) = A A B 7. Penyerapan: : A (AA (A B) = A A (A B) = A 29

30 Latihan. Buktikan A (B C) = (A B) B) (A C) 2. Buktikan A-(B C) = (A-B) B) (A-C) 3. Bila A B, buktikan A B B = A dan A B B = B 4. Buktikan (A B) x C = (AxC)( AxC) (BxC) 3

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