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

4. Algorithms Thinking Algorithm is the

specification for solving problems. Its

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

A proposition is a statement that is either true

or false, but not both.

Examples

Propositions 1. This class has 25 students. 2.

4812 3. 537

Not propositions 1. What time is it? 2. Read

this carefully. 3. x1 2.

- 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.

Examples

Examples

Examples

Examples

Another example If today is Friday, then 236.

Examples

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.

- 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.

Example Extend bit operations to bit strings.

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.

- Using a truth table to determine whether two

propositions are equivalent

Example 3

Example 2

- Some important
- equivalences.

Example 4

Example 5

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.

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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.

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