1 Introduction (Pengenalan) About the Lecturer: – Nama lengkap: Heru Suhartanto, Ph.D – Kantor: Ruang 1214, Gedung A, Fakultas Ilmu Komputer UI, Depok – E-mail: [email protected]– Pendidikan formal: – Sarjana Matematika UI, 1986 – Master of Science, Computer Science, University of Toronto, Canada, 1990. – Philosiphy Doctor (Ph.D), Parallel Computing, University of Queensland, Australia, 1998. Other lecturers – Achmad Nizar Hidayanto – Ade Azurat – Kasiyah M. Yunus – Dina Cahyati – Siti Aminah Materi Matrikulasi Matematika – pengenalan (lihat Outline), sebagian diberikan dalam text bahasa Inggris. Materi: http://telaga.cs.ui.ac.id/WebKuliah/Matrikulasi/math/
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1 Introduction (Pengenalan) n About the Lecturer: –Nama lengkap: Heru Suhartanto, Ph.D –Kantor: Ruang 1214, Gedung A, Fakultas Ilmu Komputer UI, Depok.
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Introduction (Pengenalan) About the Lecturer:– Nama lengkap: Heru Suhartanto, Ph.D– Kantor: Ruang 1214, Gedung A, Fakultas Ilmu Komputer UI, Depok– E-mail: [email protected]– Pendidikan formal:
– Sarjana Matematika UI, 1986– Master of Science, Computer Science, University of Toronto, Canada, 1990.– Philosiphy Doctor (Ph.D), Parallel Computing, University of Queensland,
Australia, 1998. Other lecturers
– Achmad Nizar Hidayanto– Ade Azurat– Kasiyah M. Yunus– Dina Cahyati– Siti Aminah
Materi Matrikulasi Matematika – pengenalan (lihat Outline), sebagian diberikan dalam text bahasa Inggris.
10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian
product of A and B is denoted as A x B.
– It obeys the following axiom:(x,y) A B iff xA yB
– We can also write:A B = { (x,y) | xA yB}
Examples:– {1,2} x {2,3} = {(1,2),(1,3),(2,2),(2,3)}
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10. Cartesian Product
10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian
product of A and B is denoted as A x B.
– It obeys the following axiom:(x,y) A B iff xA yB
– We can also write:A B = { (x,y) | xA yB}
Examples:– {1,2,3} x {a,b}
= {(1,a),(2,a),(3,a), (1,b),(2,b),(3,b)}
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10. Cartesian Product
10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian
product of A and B is denoted as A x B.
– It obeys the following axiom:(x,y) A B iff xA yB
– We can also write:A B = { (x,y) | xA yB}
Examples:– {{1},2,{3,4}} x {a,b}
= { ({1},a), (2,a), ({3,4},a),
({1},b), (2,b), ({3,4},b)}
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10. Cartesian Product
10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian
product of A and B is denoted as A x B.
– It obeys the following axiom:(x,y) A B iff xA yB
– We can also write:A B = { (x,y) | xA yB}
Q: {1,2} x {} = ? A: {}
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10. Cartesian Product
10.2 Definition (Generalised definition of cartesian product):
Given sets A1,…,An, A1 A2 … An is the set of all ordered n-tuples (x1,…,xn) where x1A1 x2A2 … xnAn
Examples:{1,2} x {2,3} x {a,b}
= {(1,2,a), (1,2,b), (1,3,a), (1,3,b), (2,2,a),
(2,2,b), (2,3,a), (2,3,b)}
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10. Cartesian Product (Proofs)
10.3 Show that A x (B C) (A x B) (A x C)
Proof:
Assume (m,n) A x (B C) m A n (B C) m A (n B n C) (m A n B) (m A n C) ((m,n) A x B) ((m,n) A x C)
(m,n) (A x B) (A x C)
Therefore A x (B C) (A x B) (A x C)
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11. Disjoint Unions
11.1 Definition:a. Two sets A and B are disjoint iff they
have no elements in common. In other words, A and B are disjoint A B =
b. A1,A2,…,An are mutually disjoint iff
i,j, Ai Aj =
c. {A1,A2,…,An } is a partition of A iff
i. A = A1 A2 … An
ii. A1,A2,…,An are mutually disjoint
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11. Disjoint Unions
Partitioning a set
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11. Disjoint Unions
11.2 Example: Let Z be the set of all integers.– Let A = {n Z | n = 3k for some integer k}– Let B = {n Z | n = 3k+1 for some integer k}– Let C = {n Z | n = 3k+2 for some integer k}
A = {…,-6,-3,0,3,6,…} B = {…,-5,-2,1,4,7,…} C = {…,-4,-1,2,5,8,…} A B = A C = B C = Z = A B C Therefore {A, B, C} form a partition of Z.
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12. Summary: Axiomatic Definitions Subset: A B iff x, xA xB Set Equality: A B iff A B B A Strict Subset: A B iff A B A B Union: x, x (A B) iff xA xB Intersection: x, x (A B) iff xA xB Difference: x, x (A B) iff xA xB Complement: x, x Ac iff xA Empty Set: (x, x {}) …or…(x, x A) A = {} Universal Set: (x, x U) …or …(x, x A) A = U Power Set: S, (S A) (S P(A)) Tuple Equality: (x1,…,xn) = (y1,…,ym) iff
n=m x1 = y1 x2=y2 … xn=yn
Cartesian Prod:(x,y) A B iff xA yB Disjoint Union: …
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Power sets, disjoint unions, ordered pairs and Cartesian Products are used in the lectures on Relations.