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 Counting techniques Permutations and combinations Pascal’s triangle and combinatorial identities

Parity Whether an integer is even or odd. Questions: 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.

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?

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

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

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:

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?

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 …

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