Applied Discrete Structures

Glossary An Informal Glossary of Terms

An.

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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.

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

Byte.

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

Complex Number.

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

Composite Integer.

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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.

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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.

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

Data Structure.

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

Distinct.

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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.

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

Factor.

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

Iff.

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Shorthand for “if and only if”

Integer.

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Whole number, whether positive, negative or zero.

Irrational Number.

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

LaTeX.

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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.

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

Natural Numbers.

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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.

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A number that is either positive or zero.

Odd Integer.

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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.

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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.

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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.

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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.

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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.

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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.

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

Real Number.

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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.

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

Stack.

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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.

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

Term.

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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.

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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.

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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.