Applied Discrete Structures

Glossary An Informal Glossary of Terms

An.

When referring to “an entity” we mean that the object can be any of the elements is some set. For example, if you say that \(n\) is an integer, it could be any integer.

Bit.

The smallest unit of computer memory, normally represented as a 0 or 1.

Byte.

A basic unit of computer memory containing eight Bits, normally modeled as a sequence of eight 0’s and 1’s.

Complex Number.

A number of the form \(a + b \boldsymbol{i}\text{,}\) where \(a\) and \(b\) are real numbers and \(\boldsymbol{i}^2=-1\text{.}\)

Composite Integer.

A positive integer is composite if it is greater than one and is the product of two positive integers greater than one. For example, 10 (equal to \(2\cdot 5\)) is composite. Any positive integer greater than one that is not composite is Prime.

Constant.

A numerical value that is unchanging . The value might be unknown and it still may be represented with a symbol. For example if we are discussing the process of sorting a file of \(N\) numbers, \(N\) is considered a constant with respect to the sorting algorithm. Constants can become variables though. If we have designed a sorting algorithm, and want to analyze its efficiency, we would consider \(N\) to be a variable.

Creative Commons.

An organization which has created several open licenses for creative works such as Applied Discrete Structures.

Data Structure.

A format for organizing, processing, retrieving and storing data.

Distinct.

Two entities are distinct if they are not the same. For example, any two student ID numbers at a school should be distinct. If not, confusion could ensue. See also Unique.

Even Integer.

Any Integer that is equal to two times an integer. That includes 0, since \(0 = 2 \cdot 0\text{.}\)

Factor.

If an algebraic expression is the product of several expressions, each of those expressions is a factor.

Iff.

Shorthand for “if and only if”

Integer.

Whole number, whether positive, negative or zero.

Irrational Number.

A number that is not equal to any fraction. \(\sqrt{2}\) is one we prove to be irrational in the book.

LaTeX.

A markup language used for books and papers with lots of mathematics, which is built on TeX. PreTeXt uses LaTeX as an intermediate format to produce PDF and print output.

Multiples.

Multiples of a number \(c\) are \(\dots, -3c,—2c,-c,0,c,2c,3c,\dots\)

Natural Numbers.

In this book, its the numbers 0,1,2,3,4,… . There isn’t 100% agreement here. Some people say its the numbers 1,2,3,4, … . We call those numbers the positive integers. The symbol we use of the natural numbers is \(\mathbb{N}\text{.}\) There is no consistent definition of positive complex numbers.

Nonnegative Number.

A number that is either positive or zero.

Odd Integer.

An integer \(n\) is odd if there exists an integer \(k\) so that \(n = 2k + 1\text{.}\) Any integer that is not even is odd.

Positive Number.

A positive number is a number that is greater than zero. Normally visualized as being to the right of zero on a conventional number line. The set of positive integers is denoted \(\mathbb{P}\text{.}\) The sets of positive rational and real numbers are denoted \(\mathbb{Q}^+\) and \(\mathbb{R}?^+\text{,}\) respectively

Powers.

Powers of a nonzero number \(c\) are \(\dots, c^{-3},c^{-2},c^{-1},1,c^1,c^2,c^3,\dots\text{.}\) Recall that \(c^0=1\text{.}\)

PreTeXt.

An authoring and publishing system for authors of textbooks, research articles, and monographs, especially in STEM disciplines. Applied Discrete Structures is produced using PreTeXt.

Prime.

A positive integer that is divisible by exactly two positive integers, itself and \(1\text{.}\) One is not prime, but \(2\) is the oddest prime because it’s even. See also Composite Integer.

Queue.

A conventional waiting line, with the first come-first serve service rule. A queue is a common Data Structure in computer science. See also Stack.

Rational Number.

Any real number that is equal to a quotient two integers, \(a/b\text{,}\) with \(b \neq 0\text{.}\)

Real Number.

For the purposes of this book, think of the numbers on a standard number line. All of the points make up the set of real numbers.

SageMath.

An open source computer algebra system for a wide range of symbolic and numerical mathematical computations. Originally named simply Sage.

Stack.

A Data Structure similar to a queue, but where the last come-first serve service rule is used. This wouldn’t be a fair waiting line rule, but it is a very useful data structure. See also Queue

Subtraction.

Subtraction is really addition of the negation of a number: \(a-b=a+(-b)\text{.}\)

Term.

If an algebraic expression is the sum of several expressions, each of those expressions is a term. For example there are three terms in the expression \(2y + x-(w+1)/2\text{.}\) Note that subtraction is considered the same as addition here.

Unique.

We say a mathematical entity is unique when there’s nothing else like it. For example, the solution, \(x=3\) to the equation \(2x+1=7\) is unique. No other number solves the equation. See also Distinct.

Variable.

A quantity whose value that can vary within a specified set. Normally represented by an algebraic symbol. For discrete variables it is customary to use the letters in the range from i to n, but this isn’t a rigid rule. Letters at the end of the alphabet are traditionally used for continuous variables.