0.1: Definitions
- Page ID
- 146869
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Our first task in understanding mathematics is to imagine "truth." We all think we know what this word means, but the "truth" I am talking about here is likely a little more stringent than most people imagine. So what does "truth" mean for us (at least, us reading this textbook)?
It means that we must ignore everything around us that is supposition and stubborn, ego-driven prejudice – none of that truly matters. We must be devoid of falsehoods and lies. There is no judgment. There is no right or wrong. There is no such thing as politics, history, religion, war, wealth, feast, famine, TikTok, or Facebook.
We should have a clean slate – a blank canvas. There is only truth, which is now bare and waiting to be filled in. We must look around ourselves as if we are newborns – as if we have no idea what anything is or what anything does. This is essential to understanding mathematics.
Now that we are floating in this void, we must form "truth." How is this done? What is our first step?
To interact with the world around us, we must devise a method to refer to things in our environment. We need a set of names to give to things. The object you are reading we shall call a book. The object holding liquid next to you we shall call a glass. The liquid inside of it we shall call tea.
On we go, defining everything around us. Someone may tell us that a table is the surface on which we are leaning our elbows. Yet another person may tell us that the thing beeping in the background is an alarm clock. In some way, our young, naive minds get filled with a dizzying list of names attributed to objects around us.
We further classify these objects into subgroups. Whenever you see a flat, vertical surface with a handle, you know it is called a door. Interestingly, we can now have a group of doors that all share the same properties but look completely different (e.g., my father's house has a maroon door, while mine is white).
Why does starting with definitions like the ones discussed above make sense? It’s best explained by example. Suppose your friend came to your house and said,
"I thought you may like to see the zart downtown so I bought a troda and put on some fofels so we can ginal."
Without the knowledge of what a zart, troda, fofel, or a ginal are, that sentence is completely useless. We can try to guess the meaning of each word, but we are likely to get something completely wrong. For anything to make sense, we first must have things defined.
You probably have already noticed that this textbook is written in such a way that you need to thoughtfully process information as you read. This is on purpose. I do not want you to just skim through and memorize my words. Instead, you should think carefully and critically about the information being presented to you.
Consistent Language
I have always had issues with words having multiple meanings. It makes life more confusing when a word has too many implications. It is like having two siblings with the same name. Who would be so cruel?1 It's bad enough having been raised with the same name as my father. I cannot recall how many times that caused confusion.
From a student's standpoint, having a word or phrase mean several different things is a complete disaster. Unfortunately, it occurs all too often. Consider the following example.
How many distinct interpretations can you think of for the word "set?"
- Answer
- Here are a few definitions I found for the word "set" - this is not an exhaustive list.
- to put into another place or location (e.g., "Set it down over there.")
- to determine or fix conclusively (e.g., "Let's set the rules for this game.")
- to establish as the highest level (e.g., "Set a new world record!")
- to ready in advance (e.g., "Is everyone set to go?")
- to locate (e.g., "The book is set in Europe.")
- to apply or start (e.g., "The vandals set fire to the building.")
- a group of objects (e.g., "This is my set of dolls.")
- a predetermined time period in a game (e.g., "They played one set of tennis.")
- a slang term used to reference a television (e.g., "Did you watch the commercial on your TV set?")
- a musician's term for a list of songs to be played (e.g., "We have a set list for the gig tonight at Old Ironsides.")
In mathematics, the word distinct is frequently used. The literal meaning of distinct is "different in nature or quality."
Example \( \PageIndex{ 1 } \) illustrates the difficulties that ambiguous language can present. The English language is full of ambiguities, which is why many second-language learners find English difficult to master. Often, ambiguous language leads to disastrous interpretations.
The following example is an actual incident experienced by my former philosophy professor at Sacramento City College.
As a student walked by his former professor, he turned to his friend and said (loud enough for the professor to hear), "There goes my old professor." Explain why the professor may take offense to this statement.
- Answer
- The student was likely pointing out to some of his friends that the professor was his previous instructor; however, the professor could have had a very sensitive ego and felt the student was calling him old. In this case, "old" has two meanings: previous and elderly.
To avoid ambiguities like these, our language, at least in mathematics, must be precise. Every definition should be scrutinized, and contrary meanings for a single term should be removed from our dictionary. Despite this, some words in mathematics will seemingly have multiple meanings. Whenever such a word is encountered in this textbook, I will clarify the supposed double meaning.
1 No offense to George Foreman and his five sons… all named George.
The Definition of... Definition
For the rest of your college mathematical career, you will be continually presented with definitions, theorems, lemmas, corollaries, axioms, and many other mathematical knick-knacks.2 Each has its place in the mathematical collage and the more familiar you are with these distinctions, the better you will do in this, and future, mathematics courses.
A definition is something that we state concerning the meaning of a word or phrase. In general, when someone defines something (by giving it a name or a "naming phrase"), there is little else you can say to refute that definition. For example, a "cell phone" is called a cell phone because someone else did not call it a "tark phone" first. Unfortunately, as mentioned previously, some definitions are ambiguous.
A chair is an object consisting of a seat, legs, back, and often arms, designed to accommodate one person in the sitting position.
Some may argue that they have a chair with no back or legs, so this definition is flawed. I completely agree. This is the problem with many definitions in the "real world" – they can be left to interpretation. A well-built definition can withstand this scrutiny, but building such a definition is complex and can lead to very long definitions if you are not careful.
The hue of the long-wave end of the visible spectrum, evoked in the human observer by radiant energy with wavelengths of approximately 630 to 750 nanometers, is defined to be the color red.
Even if you tackle the philosophical question of whether your version of red is the same as my version of red, this definition holds no ambiguity. A strong definition has this quality of precision.
It is imperative to note that definitions work both ways. Therefore, by the last definition, if we have a red light, our definition tells us it must emit wavelengths of approximately 630 to 750 nanometers. On the other hand, if we have energy in the form of light at wavelengths of approximately 630 to 750 nanometers, our definition states that it must emit the color red.
Definitions in mathematics are similar to those elsewhere in that they are statements that we accept because we must agree to some common language to communicate effectively. However, mathematical definitions differ from traditional ones in that multiple definitions for the same concept are generally not allowed.
Mathematics is not entirely devoid of multiple terms for the same definition. Whenever a definition has several different words associated with it, I will mention each word form and clarify what the most popular form is.
In the "real world," we have the luxury of being able to ask what the definition of something is a few times, and, in general, people forgive our ignorance. For example, most people would forgive you if you had to ask repeatedly for the definition of the Higgs Boson (a.k.a. the Higgs particle). It's a tricky definition to grasp.
In contrast, when taking a mathematics course, you must familiarize yourself with the definitions as soon as they are presented. The rest of the material is built upon a strong knowledge of these definitions. I will constantly use the terms we define to help you get used to mathematical definitions as quickly as possible. Moreover, I will make every effort to be consistent with my language.
In all honesty, we could spend much more time discussing the structure of a definition; however, what we have developed thus far will suffice for our needs.
2 At this point, don’t worry too much about theorems, lemmas, corollaries, and axioms. We will get to these topics in the next section.