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Discrete Mathematical Structures. . http://sist.sysu.edu.cn/~qiaohy/DiscreteMath/ qiaohy@mail.sysu.edu.cn slides contributed by Dr. Wu Xiangjun. Course Contents. Cantors Set theory, including sets, relations and functions. - PowerPoint PPT Presentation

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Discrete Mathematical Structures http://sist.sysu.edu.cn/~qiaohy/DiscreteMath/ qiaohy@mail.sysu.edu.cn slides contributed by Dr. Wu Xiangjun

Course ContentsCantors Set theory, including sets, relations and functions. Mathematical Logic, including propositions, logical operations and mathematical proofs.Graph Theory, including trees, graphs, and graph algorithms.Group theory.

Course GoalsLearn mathematical knowledge that will be used in solving problems;Learn abstraction and mathematical thinking;Learn doing mathematical proofs.

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Discrete Mathematical Structures (Fifth Edition) Bernard Kolman, Robert C. Busby and Sharon C. Ross, , 20056 Textbook Discrete Mathematical Structures Theory and Applications D. S. Malik, , 20057 Discrete Mathematics (Fifth Edition) Kenneth A. Ross, Charles R. B. Wright, , 2003() , , 2004 , , 2002 , , 1985 Reference Book

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RequirementsParticipating in Lectures actively, asking questions and taking notesReading the English text book, getting used to read English materialsFinishing homework individually before the dead line. No acceptance after the deadline.

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Grading SchemeHomework and attendance 20% Midterm 20% Final 60% Cheating may make you fail the course. We may increase the percentage of homework and attendance t0 30% or higher.

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1.1 Sets and Subsets1.2 Operations on Sets1.3 Sequences1.6 Mathematical StructuresChapter 1 Fundamentals

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quizLet S ={}, T = {, {}}Which is true? S S T T S S S S S T S T T T

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Quiz Let S ={}, T = {, {}}Which is true? S T = S S T = T S T = S S T = T S T = S T = S S T = S T =

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1.1 Sets and SubsetsA Set is any well-defined collection of objects called the elements or members of the set().1). Describe a set1.1). List the elements of the set between bracesThe set of all positive integers that are less than 4 can be written as { 1, 2, 3 }.1.2). Describe a set by statement P{ x | P(x) } is just a set which P(x) is true.Ex. { x | 0 < x < 4 }, P(x) is sentence 0 < x < 4. { y | y is a letter in the word Hello }.

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1.1 Sets and Subsets2). The order of elements of setThe order in which the elements of a set are listed is not important.{ 1, 2, 3 } = { 2, 3, 1 } = { 3, 2, 1 } = 3). Denotation for element of setWe use uppercase letters such as A, B, C to denote sets, lowercase letters such as a, b, c to denote the members of sets.(1). x is a member of set A, write x A (2). x is not a member of set A, write x A

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1.1 Sets and Subsets4). Some important setsN = { x | x 0, x is an integer }Z = { x | x is an integer } = { 0, 1, 2, 3, }Z+ = { x | x > 0, x Z }Q = { x | x is a rational number } = { x | x = a/b, a, b Z, b 0 }R = { x | x is a real number } or { } stands for empty set, it has no elements.Ex. { x | x2 = -1, x R } is .

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1.1 Sets and Subsets5). Subset and Proper Subset ()If whenever x A then x B, we say A is a subset of B, or A is contained in B. we write A B.If A is not a subset of B, we write A B.Ex. N Z Q R, but Z N.If A is a set, then A A. i.e. every set is a subset of itself.Ex. A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 5 }, C = { 1, 2, 3, 4, 5 }Then B A, B C, C A, A B, A C, C B.If B A and B A, then B is a proper subset of A(), and is denoted by B A. BAVenn diagrams

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1.1 Sets and Subsets6). Difference between and B = {A, {A}}, A is a set.A B, {A} B, {A} B and {{A}} B.A B.If A is a set, then A is true.The empty set is a subset of any set.7). Equality of two setsTwo sets A and B are equal if they have same elements, we write A = B.A = { 1, 2, 3 }, B = { x | x2 < 12, x Z+ } A=BA = B iff A B and B A.

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1.1 Sets and Subsets8). Universal set ()An universal set is a set which contains all objects for which the discussion is meaningful. U is abbreviated from Universal Set.If set A is a set in the discussion, then A is a subset of U.In Venn diagrams, set U is denoted by a rectangle.UARussells Paradox

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1.1 Sets and Subsets9). Cardinality of set (, )A set A is called finite() if it has n distinct elements, where n N.In this case, n is called the cardinality of set A, and is denoted by |A|, or # of A.If a set is not finite, it is called infinite().Ex. N, Z, R, etc.

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1.1 Sets and Subsets10). Power set ()If A is a set, the set of all subset of A is called the power set of A, and is denoted by P(A).Ex. A = { 1, 2, 3 }P(A) = { , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }|P(A)| = 8.|P(A)| = 2|A|.

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1.2 Operations on Sets1). UnionThe union of A and B is a set which contains all elements of A or B, we write AB.AB = { x | x A or x B }Ex. A = { 1, 2, 3, 4, 5 }, B = { 3, 5, 6, 7, 8 } AB = { 1, 2, 3, 4, 5, 6, 7, 8 }AB

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1.2 Operations on Sets2). IntersectionThe intersection of A and B is a set which contains all elements of A and B, we write AB.AB = { x | x A and x B }Ex. A = { 1, 2, 3, 4, 5 }, B = { 3, 5, 6, 7, 8 } AB = { 3, 5 }If set A and B have no common elements, they are called disjoint sets().ABDisjoint Sets

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1.2 Operations on SetsABC = { x | x A or x B or x C }i=1..nAi = A1A2 AnABC = { x | x A and x B and x C }i=1..nAi = A1A2 AnABABCCABABCC

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1.2 Operations on Sets3). Complement of B with respect to AIf A and B are two sets, The complement of B with respect to A is defined the set of all elements that belong to A but not to B, and is denoted by A - B.A - B = { x | x A and x B }Ex. A = { 1, 2, 3, 4, 5 }, B = { 3, 5, 6, 7, 8 } A - B = { 1, 2, 4 }AA - BBB - A1

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1.2 Operations on Sets4). Complement of AAIf U is a universal set containing A, U-A is the comple-ment of A, and is denoted by A.A = U-A = { x | x U and x A }Ex. A = { 1, 2, 3, 4, 5 }, U = Z+ A = { x | x is an integer and x > 5 }UAA

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1.2 Operations on Sets5). Symmetric difference A and B are two sets, their symmetric difference is the set of all elements that belong to A or to B, but not to A and B, it is denoted by AB.AB = { x | (x A and x B) or (x B and x A) } = (A B)(B A) = (AB) (AB)Ex. A = { 1, 2, 3, 4, 5 }, B = { 3, 5, 7, 9 } AB = { 1, 2, 4, 7, 9 }AB

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1.2 Operations on Sets6) Algebraic properties of set operations(1) Commutative Properties AB = BAAB = BA(2) Associative Properties (AB)C = A(BC) (AB)C = A(BC)(3) Distributive Properties A(BC) = (AB)(AC) A(BC) = (AB)(AC)(4) Idempotent Properties AA = AAA = A

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1.2 Operations on Sets6) Algebraic properties of set operations(5) Properties of the Complement A = A AA = UAA = = UU = AB = AB AB = AB(6) Properties of a Universal Set AU = UAU = A(7) Properties of the Empty Set A = AA = De Morgans Laws

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1.2 Operations on Sets6) Algebraic properties of set operationsDe Morgans law: AB = ABProof:(1). Suppose that x AB.Then x AB, so x A and x B.x A, x B, so x AB.Thus AB AB.(2). Suppose that x AB.Then x A, x B, so, x A, x B.Thus, x AB, x AB.We have that AB AB.Thus, we hold that AB = AB.A common style of proof for statements of sets is to choose an element in one of the sets.

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QuizWhich of the following is true? Prove or disprove it. A B = A B A B = A B A (B C) = A B C A (B C) = (A B )(A C) A (B C) = (A B) (A C)

Hints: Use Venn diagrams and operation laws.

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The addition principleQuestions: How many of you can program in C# or Java?How many of you can program in Java?How many of you can program in C#?How many can program in both C# and Java?

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1.2 Operations on Sets7) The addition principleTheorem 2. If A and B are finite sets, then |AB| = |A| + |B| - |AB|Ex. A = { a, b, c, d, e }, B = { c, e, f, h, k, m }, |AB| = ? AB = { c, e }, |AB| = 2. |AB| = |A| + |B| - |AB| = 5 + 6 2 = 9.Theorem 3. If A, B and C are finite sets, then |ABC | = |A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|Ex. A = { a, b, c, d, e }, B = { a, b, e, g, h }, C = { b, d, e, g, h, k, m, n }, |ABC | = ?

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1.2 Operations on Sets 11000(11000), 56, 8. S = { x | xZ1 x 1000 }A = { x | xSx5 }B = { x | xSx6 }C = { x | xSx8 }|ABC| =| ABC| = 1000 - | ABC| |A| = 1000/5 = 200, |B| = 1000/6 = 166, |C| = 1000/8 = 125|AB| = 1000/lcm(5,6) = 33, |AC| = 1000/lcm(5,8) = 25|BC| = 1000/lcm(6,8) = 41|ABC| = 1000/lcm(5,6,8) = 8| ABC| = 400|ABC| = 1000-400=600

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1.2 Operations on Sets 24.: : 13, 5, 109;2;4., (). E, F, GJ.x, y1, y2y3..y224-xy1x4-x4-xy35-2JEGF

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1.2 Operations on Sets 24.: : 13, 5, 109;2;4., (). E, F, GJ.x, y1, y2y3..:y1 + 2(4-x) + x + 2 = 13y2 + 2(4-x) + x = 9y3 + 2(4-x) + x = 10y1 + y2 + y3 + 3(4-x) + x = 19: x = 1, y1 = 4, y2 = 3, y3 = 3.y224-xy1x4-x4-xy35-2JEGF

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1.2 Operations on Sets8) The characteristic functionThe characteristic function fA of A is defined for each x U as follows:fA(x) = 1x A0x ATheorem 4. Characteristic function of subsets satisfy the following properties:(a). fAB = fAfB, fAB(x) = fA(x)fB(x) for all x.(b). fAB = fA + fB - fAfB, fAB(x) = fA(x)+fB(x)-fA(x)fB(x) for all x.(c). fAB = fA + fB - 2fAfB, fAB(x) = fA(x)+fB(x)-2fA(x)fB(x) for all x.Example: U = N, A = {0,2,4,}, fA, fN, f ?

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1.3 Sequences1.4 Division in the Integers1.5 MatricesWhich of the following sets has more elements?N {x| x = 2n, nN} {p| p is a prime} Q R

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Computer Representations of SetsExample 1: the union functionExample 2: Message flood (soj.me/1443).

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1.6 Mathematical StructuresA collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system. is a mathematical structure, where Sets is set of sets on some universe , , and ~ are operations of set: union, intersection and complement., are a mathematical structure too.(1). An