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Diterbitkan olehAxel Lina Telah diubah "10 tahun yang lalu
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Prof. Busch - LSU1 Mathematical Preliminaries
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Prof. Busch - LSU2 Mathematical Preliminaries Sets Functions Relations Graphs Proof Techniques
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Prof. Busch - LSU3 A set is a collection of elements SETS We write
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Prof. Busch - LSU4 Set Representations C = { a, b, c, d, e, f, g, h, i, j, k } C = { a, b, …, k } S = { 2, 4, 6, … } S = { j : j > 0, and j = 2k for some k>0 } S = { j : j is nonnegative and even } finite set infinite set
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Prof. Busch - LSU5 A = { 1, 2, 3, 4, 5 } Universal Set: all possible elements U = { 1, …, 10 } 1 2 3 4 5 A U 6 7 8 9 10
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Prof. Busch - LSU6 Set Operations A = { 1, 2, 3 } B = { 2, 3, 4, 5} Union A U B = { 1, 2, 3, 4, 5 } Intersection A B = { 2, 3 } Difference A - B = { 1 } B - A = { 4, 5 } U A B 2 3 1 4 5 2 3 1 Venn diagrams
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Prof. Busch - LSU7 A Complement Universal set = {1, …, 7} A = { 1, 2, 3 } A = { 4, 5, 6, 7} 1 2 3 4 5 6 7 A A = A
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Prof. Busch - LSU8 0 2 4 6 1 3 5 7 even { even integers } = { odd integers } odd Integers
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Prof. Busch - LSU9 DeMorgan’s Laws A U B = A B U A B = A U B U Proof it!
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Prof. Busch - LSU10 Empty, Null Set: = { } S U = S S = S - = S - S = U = Universal Set
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Prof. Busch - LSU11 Subset A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 } A B U Proper Subset:A B U A B
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Prof. Busch - LSU12 Disjoint Sets A = { 1, 2, 3 } B = { 5, 6} A B = U AB
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Prof. Busch - LSU13 Set Cardinality For finite sets A = { 2, 5, 7 } |A| = 3 (set size)
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Prof. Busch - LSU14 Powersets A powerset is a set of sets Powerset of S = the set of all the subsets of S S = { a, b, c } 2 S = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } Observation: | 2 S | = 2 |S| ( 8 = 2 3 )
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Prof. Busch - LSU15 Cartesian Product A = { 2, 4 } B = { 2, 3, 5 } A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) } |A X B| = |A| |B| Generalizes to more than two sets A X B X … X Z
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Prof. Busch - LSU16 FUNCTIONS domain 1 2 3 a b c range f : A -> B A B If A = domain then f is a total function otherwise f is a partial function f(1) = a 4 5
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Prof. Busch - LSU17 RELATIONS R = {(x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ), …} x i R y i e. g. if R = ‘>’: 2 > 1, 3 > 2, 3 > 1
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Prof. Busch - LSU18 Equivalence Relations Reflexive: x R x Symmetric: x R y y R x Transitive: x R y and y R z x R z Example: R = ‘=‘ x = x x = y y = x x = y and y = z x = z
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Prof. Busch - LSU19 Equivalence Classes For equivalence relation R equivalence class of x = {y : x R y} Example: R = { (1, 1), (2, 2), (1, 2), (2, 1), (3, 3), (4, 4), (3, 4), (4, 3) } Equivalence class of 1 = {1, 2} Equivalence class of 3 = {3, 4}
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Prof. Busch - LSU20 GRAPHS A directed graph Nodes (Vertices) V = { a, b, c, d, e } Edges E = { (a,b), (b,c), (b,e),(c,a), (c,e), (d,c), (e,b), (e,d) } node edge a b c d e
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Prof. Busch - LSU21 Labeled Graph a b c d e 1 3 5 6 2 6 2
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Prof. Busch - LSU22 Walk a b c d e Walk is a sequence of adjacent edges (e, d), (d, c), (c, a)
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Prof. Busch - LSU23 Path a b c d e Path is a walk where no edge is repeated Simple path: no node is repeated
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Prof. Busch - LSU24 Cycle a b c d e 1 2 3 Cycle: a walk from a node (base) to itself Simple cycle: only the base node is repeated base
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Prof. Busch - LSU25 Euler Tour a b c d e 1 2 3 4 5 6 7 8 base A cycle that contains each edge once
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Prof. Busch - LSU26 Hamiltonian Cycle a b c d e 1 2 3 4 5 base A simple cycle that contains all nodes
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Prof. Busch - LSU27 Finding All Simple Paths a b c d e origin
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Prof. Busch - LSU28 (c, a) (c, e) Step 1 a b c d e origin
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Prof. Busch - LSU29 (c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d) Step 2 a b c d e origin
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Prof. Busch - LSU30 Step 3 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, e) (c, e), (e, b) (c, e), (e, d)
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Prof. Busch - LSU31 Step 4 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, a), (a, b), (b, e), (e,d) (c, e) (c, e), (e, b) (c, e), (e, d)
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Prof. Busch - LSU32 Trees root leaf parent child Trees have no cycles
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Prof. Busch - LSU33 root leaf Level 0 Level 1 Level 2 Level 3 Height 3
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Prof. Busch - LSU34 Binary Trees
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Prof. Busch - LSU35 PROOF TECHNIQUES Proof by induction Proof by contradiction
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Prof. Busch - LSU36 Induction We have statements P 1, P 2, P 3, … If we know for some b that P 1, P 2, …, P b are true for any k >= b that P 1, P 2, …, P k imply P k+1 Then Every P i is true
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Prof. Busch - LSU37 Proof by Induction Inductive basis Find P 1, P 2, …, P b which are true Inductive hypothesis Let’s assume P 1, P 2, …, P k are true, for any k >= b Inductive step Show that P k+1 is true
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Prof. Busch - LSU38 Example Theorem: A binary tree of height n has at most 2 n leaves. Proof by induction: let L(i) be the maximum number of leaves of any subtree at height i
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Prof. Busch - LSU39 We want to show: L(i) <= 2 i Inductive basis L(0) = 1 (the root node) Inductive hypothesis Let’s assume L(i) <= 2 i for all i = 0, 1, …, k Induction step we need to show that L(k + 1) <= 2 k+1
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Prof. Busch - LSU40 Induction Step From Inductive hypothesis: L(k) <= 2 k height k k+1
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Prof. Busch - LSU41 L(k) <= 2 k L(k+1) <= 2 * L(k) <= 2 * 2 k = 2 k+1 Induction Step height k k+1 (we add at most two nodes for every leaf of level k)
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Prof. Busch - LSU42 Proof by Contradiction We want to prove that a statement P is true we assume that P is false then we arrive at an incorrect conclusion therefore, statement P must be true
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43 Example Theorem: Jika n 2 genap maka n genap Proof: Dengan contradiction kita asumsikan jika n 2 genap maka n ganjil We will show that this is impossible
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44 Bukti: Diketahui n 2 genap (kita asumsikan benar) Andaikan n ganjil (negasinya) Karena n ganjil maka n bisa ditulis dengan bentuk n = 2m + 1 dengan m bilangan bulat
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45 Diperoleh n 2 = (2m + 1)2 n 2 = 4m 2 + 4m +1 n 2 = 2(2m 2 + 2m) + 1 Itu berarti n 2 = 2(2m 2 + 2m) + 1 ganjil. Kontradiksi dengan asumsi “n 2 genap” karena terjadi kontradiksi maka dapat disimpulkan jika n 2 genap maka n genap
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Prof. Busch - LSU46 Example Theorem: is not rational Proof: Assume by contradiction that it is rational = n/m n and m have no common factors We will show that this is impossible
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Prof. Busch - LSU47 = n/m 2 m 2 = n 2 Therefore, n 2 is even n is even n = 2 k 2 m 2 = 4k 2 m 2 = 2k 2 m is even m = 2 p Thus, m and n have common factor 2 Contradiction!
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Latihan 48
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Latihan Buatlah notasi bagi set Z yang mempunyai anggota {0,1,2,3,4} –N mewakili bilangan natural (bilangan asli) Buatlah notasi bagi set Y yang merupakan bilangan ganjil Buatlah notasi bagi set yang terdiri dari 2 pasang (2-tuples) bilangan natural dengan syarat nilai elemen pertama lebih kecil dari elemen kedua pada tuples 49
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