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Discrete Mathematics for Computing (Draft)
Christine Caples, Adam Gilbert, Melanie Fraser
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Front Matter
Colophon
About This Text
1
Base Representations for Number Systems
1.1
Introduction
1.2
Converting from Non-Standard Bases into Base-10
1.2.1
Numbers in Non-Standard Bases
1.2.2
Common Number Systems
1.2.3
Conclusion
1.3
Converting from Base-10 to Non-Standard Bases
1.3.1
Developing a Strategy
1.3.2
Let’s Try It!
1.3.3
Converting to Hexadecimal
1.3.4
Conclusion
1.4
Properties of Binary, Octal, and Hexadecimal Systems
1.4.1
Properties of Binary Numbers
1.4.2
Connecting Binary to Octal and Hexadecimal
1.4.3
Conclusion
1.5
Converting Between Arbitrary Bases
2
Sets and Set Operations
2.1
Foundations for Working with Sets
2.2
Operations with Sets
3
Symbolic Logic, Boolean Algebra, and Flow Control
3.1
Statements and Symbolic Logic
3.1.1
Statements
3.1.2
Symbolic Logic
3.1.3
Summary
3.2
Compound Statements, Truth Tables, and Logical Equivalence
3.2.1
Truth Tables
3.2.2
Compound Statements and Truth Tables
3.2.3
Logical Equivalence
3.3
Boolean Algebra
3.3.1
Properties in Boolean Algebra
3.4
Logic for Flow Control
3.4.1
Conditional Statements in Code
3.4.2
Conditionals with Compound Conditions
3.4.3
Looping with
for
and
while
3.4.4
Putting it all together
3.4.5
Conclusion
3.5
Building a Base-Conversion Function
4
Counting
4.1
Basic Counting Techniques
4.1.1
Counting Outcomes of Atomic Processes
4.1.2
Near-Atomic Processes: One
or
the Other(s)
4.2
The Fundamental Principle of Counting
4.2.1
Motivation
4.2.2
The Fundamental Principle of Counting (and Examples)
4.3
Arrangements and Permutations
4.3.1
Motivation
4.3.2
Interactive Examples Involving Permutations
4.4
Selections and Binomial Coefficients
4.4.1
Motivation
4.4.2
Formalizing a Method
4.4.3
Examples to Try
4.5
Complexity and Big-O Notation
5
Functions
6
Probability
6.1
Discrete Probability Basics
6.1.1
Probability and
Atomic
Processes
6.1.2
Compound Events and their Probabilities
6.2
Consecutive Events: Independence or Dependence
6.2.1
Consecutive Events, Independence/Dependence, and Probabilities
6.3
Discrete Probability and Counting
6.3.1
Motivation
7
Arrays
7.1
One-Dimensional Arrays
7.1.1
Addition and Scalar Multiplication
7.1.2
Inner-products
7.2
Two-Dimensional Arrays (Matrices)
7.2.1
What is a two-dimensional array?
7.2.2
Matrix Addition/Subtraction
7.2.3
Scalar Multiplication
7.2.4
Matrix Multiplication
7.2.5
Outer-products of one-dimensional arrays
7.2.6
Conclusions
7.3
Multidimensional Arrays
8
Graph Theory
Chapter
1
Base Representations for Number Systems
We’ll begin our study of Discrete Mathematics for Computing by thinking more about the ways we have chosen to represent numbers. Additionally, we’ll consider methods which are much more natural for computers to encode numbers.
1.1
Introduction
1.2
Converting from Non-Standard Bases into Base-10
1.3
Converting from Base-10 to Non-Standard Bases
1.4
Properties of Binary, Octal, and Hexadecimal Systems
1.5
Converting Between Arbitrary Bases
