## Table of Contents

1. Logic

1.1 Propositions and logical operations

1.2 Evaluating compound propositions

1.3 Conditional statements

1.4 Logical equivalence

1.5 Laws of propositional logic

1.6 Predicates and quantifiers

1.7 Quantified Statements

1.8 De Morgan’s law for quantified statements

1.9 Nested quantifiers

1.10 More nested quantified statements

1.11 Logical reasoning

1.12 Rules of inference with propositions

1.13 Rules of inference with quantifiers

2. Proofs

2.1 Mathematical definitions

2.2 Introduction to proofs

2.3 Best practices and common errors in proofs

2.4 Writing direct proofs

2.5 Proof by contrapositive

2.6 Proof by contradiction

2.7 Proof by cases

3. Sets

3.1 Sets and subsets

3.2 Set of sets

3.3 Union and intersection

3.4 More set operations

3.5 Set identities

3.6 Cartesian products

3.7 Partitions

4. Functions

4.1 Definition of functions

4.2 Floor and ceiling functions

4.3 Properties of Functions

4.4 The Inverse of a function

4.5 Composition of functions

4.6 Logarithms and exponents

5. Boolean Algebra

5.1 An introduction to Boolean algebra

5.2 Boolean functions

5.3 Disjunctive and conjunctive normal form

5.4 Functional completeness

5.5 Boolean satisfiability

5.6 Gates and circuits

6. Relations / Digraphs

6.1 Introduction to binary relations

6.2 Properties of binary relations

6.3 Directed graphs, paths, and cycles

6.4 Composition of relations

6.5 Graph powers and the transitive closure

6.6 Matrix multiplication and graph powers

6.7 Partial orders

6.8 Strict orders and directed acyclic graphs

6.9 Equivalence relations

6.10 N-ary relations and relational databases

7. Computation

7.1 An introduction to algorithms

7.2 Asymptotic growth of functions

7.3 Analysis of algorithms

7.4 Finite state machines

7.5 Turing machines

7.6 Decision problems and languages

8. Induction And Recursion

8.1 Sequences

8.2 Recurrence relations

8.3 Summations

8.4 Mathematical induction

8.5 More inductive proofs

8.6 Strong induction and well-ordering

8.7 Loop invariants

8.8 Recursive definitions

8.9 Structural induction

8.10 Recursive algorithms

8.11 Induction and recursive algorithms

8.12 Analyzing the time complexity of recursive algorithms

8.13 Divide-and-conquer algorithms: Introduction and mergesort

8.14 Divide-and-conquer algorithms: Binary search

8.15 Solving linear homogeneous recurrence relations

8.16 Solving linear non-homogeneous recurrence relations

8.17 Divide-and-conquer recurrence relations

9. Integer Properties

9.1 The Division Algorithm

9.2 Modular arithmetic

9.3 Prime factorizations

9.4 Factoring and primality testing

9.5 Greatest common divisor and Euclid’s algorithm

9.6 Number representation

9.7 Fast exponentiation

9.8 Introduction to cryptography

9.9 The RSA cryptosystem

10. Introduction To Counting

10.1 Sum and product rules

10.2 The bijection rule

10.3 The generalized product rule

10.4 Counting permutations

10.5 Counting subsets

10.6 Subset and permutation examples

10.7 Counting by complement

10.8 Permutations with repetitions

10.9 Counting multisets

10.10 Assignment problems: Balls in bins

10.11 Inclusion-exclusion principle

10.12 Counting problem examples

11. Advanced Counting

11.1 Generating permutations and combinations

11.2 Binomial coefficients and combinatorial identities

11.3 The pigeonhole principle

11.4 Generating functions

12. Discrete Probability

12.1 Probability of an event

12.2 Unions and complements of events

12.3 Conditional probability and independence

12.4 Bayes’ Theorem

12.5 Random variables

12.6 Expectation of a random variable

12.7 Linearity of expectations

12.8 Bernoulli trials and the binomial distribution

13. Graphs

13.1 Introduction to graphs

13.2 Graph representations

13.3 Graph isomorphism

13.4 Walks, trails, circuits, paths, and cycles

13.5 Graph connectivity

13.6 Euler circuits and trails

13.7 Hamiltonian cycles and paths

13.8 Planar graphs

13.9 Graph coloring

14. Trees

14.1 Introduction to trees

14.2 Tree application examples

14.3 Properties of trees

14.4 Tree traversals

14.5 Spanning trees and graph traversals

14.6 Minimum spanning trees

## What You’ll Find In This zyBook:

### More action with less text.

- ~750 participation activities: Questions, animations, tools
- Exceptionally visual presentations: Animations of normally hard DM concepts
- Seamlessly integrated auto-generated and auto-graded challenge activities
- Includes hundreds of end-of-section exercises

## The zyBooks Approach

### Less text doesn’t mean less learning.

A visually animated interactive introduction to discrete mathematics. This zyBook demonstrates how to translate English descriptions of everyday scenarios into precise mathematical statements that can then be used for formal analysis. Applications are described so that students get a feel for how the mathematical tools they are learning will be applied later in their studies. The material is formed from years of experience teaching discrete math to undergraduates and contains explanations of many common questions and misconceptions that students have about this material. Foundational topics provide a pathway to more advanced study in computer science. Includes hundreds of end-of-section exercises with solutions for instructors, plus numerous auto-generated, auto-graded exercises.

## Authors

**Sandy Irani**

*Professor of Information and Computer Science, University of California, Irvine*