Discrete Mathematics Goals of a Discrete Mathematics Learn how to think mathematically 1. Mathematical Reasoning Foundation for discussions of methods of proof 2. Combinatorial Analysis The method for counting or enumerating objects 3. Discrete Structure Abstract mathematical structures used to represent discrete objects and relationship between them What will we learn from Discrete Mathematics

4. Algorithms Thinking Algorithm is the specification for solving problems. It’s design and analysis is a mathematical activity. 5. Application and Modeling Discrete Math has applications to most area of study. Modeling with it is an extremely important problem-solving skill. How to learn Discrete Mathematics? Do as many exercises as you possibly can !

Chapter 1 The Foundations: Logic, Sets, and Functions Rules of logic specify the precise meaning of mathematics statements. Sets are collections of objects. A function sets up a special relation between two sets. 1.1 Logic Propositions A proposition is a statement that is either true or false, but not both. Propositions 1. This class has 25 students = =7 Not propositions 1. What time is it? 2. Read this carefully. 3. x+1= 2. Examples

Definition 1. Let p be a proposition. The statement “It is not the case of p” is a proposition, called the negation of p and denoted by We let propositions be represented as p,q,r,s,…. The value of a proposition is either T(true) or F(false). p: Toronto is the capital of Canada. Examples Table 1. The Truth Table for the negation of a proposition p TFTF FTFT called connectives

Definition 2. Let p and q be proposition ｓ.The proposition “p and q”, denoted by, is the proposition that is true when both p and q are true and is false otherwise. The proposition is called the conjunction of p and q. Table 2. The Truth Table for the conjunction of two propositions p q T T F F T F TFFFTFFF Examples

Definition 3. Let p and q be proposition ｓ.The proposition “p or q”, denoted by, is the proposition that is false when both p and q are false and is true otherwise. The proposition is called the disjunction of p and q. Table 3. The Truth Table for the disjunction of two propositions p q T T F F T F TTTFTTTF Examples

Definition 4. Let p and q be proposition ｓ.The exclusive of p and q, denoted by, is the proposition that is true when exactly one of p and q is true and it is false otherwise. Table 4. The Truth Table for the exclusive or of two propositions p q T T F F T F FTTFFTTF Examples

Definition 5. Let p and q be proposition ｓ.The implication is the proposition that is false when p is true and q is false and true otherwise ， where p is called hypothesis and q is called the conclusion. Table 5. The Truth Table for the implication p q T T F F T F TFTTTFTT Examples “If p, then q” or “ p implies q”. Another example: If today is Friday, then 2+3=6.

Definition 6. Let p and q be proposition ｓ.The biconditional is the proposition that is true when p and q have the same truth values and is false otherwise. Table 6. The Truth Table for the biconditional p q T T F F T F TFFTTFFT Examples “p if and only if q”.

Translating English Sentences into Logical Expressions Example 1 You can access the Internet from campus only if you are a computer science major or you are not a freshman. a. You can access the Internet from campus. c. You are a computer science major. f. You are freshman. The sentence can be represented as Example 2 You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old. q. You can ride the roller coaster. r. You are under 4 feet tall. s. You are older than 16 years old. The sentence can be represented as

Logic and Bit Operations A bit has two values: 1(true) and 0(false). A variable is called a Boolean variable if its value is either true or false. Bit operations are written to be AND, OR and XOR in programming languages. Table 7. Table for the bit operations OR,AND and XOR x0011x0011 y0101y Example Extend bit operations to bit strings bitwise OR bitwise AND bitwise XOR

1.2 Propositional Equivalences Definition 1. A tautology is a compound proposition that is always true no matter what the values of the propositions that occur in it. A contradiction is a compound proposition that is always false ． A contingency is a proposition that is neither a tautology nor a contradiction. Table 1. Examples of a Tautology and a Contradiction. pTFpTF FTFT TTTT FFFF a contradiction a tautology Example 1.

Logic Equivalences Definition 2. The proposition p and q are called logically equivalent if is a tautology. The notation denotes that p and q are logically equivalent. Using a truth table to determine whether two propositions are equivalent Example 2 Example 3 equivalent

Some important equivalences.

Example 4 Example 5

1.3 Predicates and Quantifiers Propositional function A statement involving a variable x is P(x) is said to be a propositional function if x is a variable and P(x) becomes a proposition when a value has been assigned to x. In general, a statement involving the n variables Example 1 Let P(x) denote the statement “x>3”. What are the truth values of P(4) and P(2)? Example 2 Let Q(x,y) denote the statement “x=y+3”. What are the truth values of the propositions Q(1,2) and Q(3,0)?

Quantifiers The universal quantification of P(x),denoted as is the proposition “P(x) is true for all values of x in the universe of discourse.” Solution P(x): x has studied calculus. S(x): x is in this class. The statement can be expressed as universal quantifier Example 3 Express the statement “Every student in this class has studied calculus.

existence quantifier The existential quantification of P(x),denoted as is the proposition “There exists an element x in the universe of discourse such that P(x) is true.”

Solution: Every student in your school has a computer or has a friend who has a computer. Solution: There is a student none of whose friends are also friends with each other.

Translating Sentences into Logical Expressions Example 10 Express the statements “Some student in this class has visited Mexico” and “every student in this class has visited either Canada or Mexico using quantifiers. Example 11 Express the statement “Everyone has exactly one best friend” as a logical expression.

Example 12 Express the statement “There is a woman who has taken a flight on every airline in the world. Negations: the negation of quantified expressions. Example 14 There is a student in the class who has taken a course in calculus. Example 13 Every student in the class has taken a course in calculus.