Discrete Mathematics

1. Discrete Mathematics • Goals of a Discrete Mathematics Learn how to think mathematically • What will we learn from Discrete Mathematics 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

2. 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 !

3. Propositions Chapter 1 The Foundations: Logic, Sets, and Functions • Rules of logicspecify the precise meaning of • mathematics statements. • Sets are collections of objects. • A function sets up a special relation between two sets. 1.1 Logic A proposition is a statement that is either true or false, but not both. Examples Propositions 1. This class has 25 students. 2. 4+8=12 3. 5+3=7 Not propositions 1. What time is it? 2. Read this carefully. 3. x+1= 2.

4. 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 called connectives Table 1. The Truth Table for the negation of a proposition p T F F T • We let propositions be represented as p,q,r,s,…. • The value of a proposition is either T(true) or F(false). Examples p: Toronto is the capital of Canada.

5. Definition 2. Let p and q be propositionｓ.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 T F F T F F T F F F Examples

6. Definition 3. Let p and q be propositionｓ.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 T F F T F F T T T F Examples

7. Definition 4. Let p and q be propositionｓ.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 T F F T F F F T T F Examples

8. Definition 5. Let p and q be propositionｓ.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. “If p, then q” or “ p implies q”. Table 5. The Truth Table for the implication p q T T T F F T F F T F T T Examples Another example: If today is Friday, then 2+3=6.

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

10. The sentence can be represented as The sentence can be represented as 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. 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.

11. Logic and Bit Operations Table 7. Table for the bit operations OR,AND and XOR x 0 0 1 1 y 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 01 1011 0110 11 0001 1101 11 1011 1111 bitwise OR 01 0001 0100 bitwise AND 10 1010 1011 bitwise XOR • A bit has two values: 1(true) and 0(false). • A variable is called a Boolean variable ifits value is either true or false. • Bit operations are written to be AND, OR and XOR in programming languages. Example Extend bit operations to bit strings.

12. Table 1. Examples of a Tautology and a Contradiction. a tautology a contradiction p T F F T T T F F 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. Example 1.

13. 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. equivalent equivalent • Using a truth table to determine whether two propositions are equivalent Example 3 Example 2

14. Some important • equivalences.

15. Example 4 Example 5

16. Propositional function Example 1 Let P(x) denote the statement “x>3”. What are the truth values of P(4) and P(2)? In general, a statement involving the n variables 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)? 1.3 Predicates and Quantifiers 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.

17. 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.” universal quantifier Example 3 Express the statement “Every student in this class has studied calculus. Solution P(x): x has studied calculus. S(x): x is in this class. The statement can be expressed as

18. 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.” existence quantifier

19. 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.

20. 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.

21. 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 13 Every student in the class has taken a course in calculus. Example 14 There is a student in the class who has taken a course in calculus.