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Introduction to Graph Theory & its Applications

Introduction to Graph Theory & its Applications. Lecture 02: Mathematical Preliminary (I). Getting Started …. This and next classes serve as a review for some elementary mathematical concepts (there should be nothing new to you) Sets Functions Parity Mathematical induction

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Introduction to Graph Theory & its Applications

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  1. Introduction to Graph Theory & its Applications Lecture 02: Mathematical Preliminary (I)

  2. Getting Started … • This and next classes serve as a review for some elementary mathematical concepts (there should be nothing new to you) • Sets • Functions • Parity • Mathematical induction • Counting techniques • Permutations and combinations • Pascal’s triangle and combinatorial identities

  3. Parity • Whether an integer is even or odd. • Even: 2n • Odd: 2n+1 • Questions: • Q1: So, is 0 even or odd? • Q2: Is parity preserved when we square an integer? • A product of integers is even at least one of the factors is even.

  4. Sets • Definition: Well-defined collection of distinct object. • A set of prime number • A set of rich people is not legitimate (why?) • meaning x being an element of A • A set is finite if we can count the number of elements it contains. Otherwise, the set is infinite. • Examples?

  5. Set • Subset: Set A is a subset of B if every element of A is also in B, • For any set A, we have • |A| is the cardinality, which is the number elements contained in A. • Power set: The set of all subset of A, • What is of A={a,b}? • If A contains n element, what is the size of

  6. Set Operations • Union : merging two sets A and B. • Intersection : set of all elements that belong to both A and B. • Both union and intersect obey • Associativity • Commutativity • Distributive law

  7. Set Operations • Universal set U: The set to which we restrict our attention • Complement of A: The set containing all elements in U but not in A. • DeMorgan’s Law:

  8. Set: Cartesian Product • : set of all ordered pairs (a,b), where and • Give A={1,2,3} and B={x,y}, find AxB • A relation from set A to B: A subset of AxB • What are the possible relations?

  9. Set: Cartesian Product • A function from A to B, : a relation in which each element of A appears as the first coordinate of precisely one ordered pair in a relation (A is the domain, and B is the range) • A function is one-to-one if the second coordinates are distinct. We must have • is onto if each element of B appears at least once. • A one-to-one and onto function is …

  10. Mathematical Induction • A powerful proof technique. It consists of two steps: • Step1 (basic step): Prove that the statement Snis true for some starting value of n • Step2 (inductive step): Assuming that Snis true for n=k, prove it is also true for n=k+1 • Prove the following statement by induction Triangular number

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