0.2: Axioms, Theorems, and Proofs
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This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Axioms
"A day without sunshine is like... well... night."
There is a strange creature in mathematics, not typically mentioned in lower division texts, called an axiom (or, in some texts, a postulate).
An axiom is a self-evident or universally recognized truth. It is accepted as true, without proof, as the basis for argument.
Like definitions, the truthfulness of any axiom is taken for granted; however, axioms do not define things – instead, they describe a fundamental, underlying quality about something. The opening quote of this section can be considered an axiom. It is not defining sunshine; it is just stating something obvious.
An axiom has the feel of something that should be justifiable or proved.a Oddly enough, though, axioms cannot be proved. They are jumping points for future logical deductions, but nothing exists to state that the axiom is true initially. The foundation of everything we know in mathematics comes from a simple set of axioms.
In the study of advanced mathematics, specifically Number Theory, axioms are further divided into logical and non-logical axioms. The remainder of this textbook, as well as most undergraduate mathematics courses, is based on non-logical axioms.
Each of the following statements is an example of an axiom.
- If I am wearing a red shirt and blue jeans, I am wearing something red.
- If I have $30 and you have $30, then we have the same amount.
- It is possible to draw a line between any two points.
The first statement in Example \( \PageIndex{ 1 } \) is almost trivial. There is no arguing that I would be wearing red because I already told you I was wearing red. The second is also obvious. Finally, it's easy to see that the third statement is a universally recognized truth. It is one of those statements that students, and most professors, would likely say, "It's true because… well… it's obvious."
When we began our exploration of numbers back in arithmetic, we were introduced to the ten fundamental axioms upon which almost all of mathematics is built. These included such catchy phrases as the "commutative property of addition," the "distributive property," and the "multiplicative identity." It takes some creative thinking to develop axioms; however, that is not the goal of this section, nor will we even approach that conversation in this textbook. For now, it's enough that you know there is an object in mathematics called an axiom.
a We will cover what a proof is in a moment.
Theorems
In essence, mathematics develops new knowledge by logical deduction from old knowledge. The new knowledge is called the conclusion, and the old knowledge is called the premise (or assumption). The following definition introduces the vehicle we will use to conclude new knowledge from old premises (or assumptions).
A theorem is a proposition that has been, or is to be, proved based on explicit assumptions.
Like many definitions, this seems a little difficult to understand without some more straightforward explanation. In layperson's terms, theorems are claims that can be proven using previous information given to us. Let's take a look at a simple theorem which is often cited in textbooks on logic.
If it is raining, then it must be cloudy.
This is a theorem because someone, namely me, is proposing that for it to rain, we must have clouds. Can this be proven? Since I am not an expert in meteorology, I can only back my claim using a common sense approach. Nonetheless, this theorem would hold true as long as my argument has no flaws.
In mathematics, the theorem is king! It is the basis of correctness, and no statement is more potent than the theorem (as long as it has been proved). I often tell my students they must only believe something if it has been proven to them. While this is admittedly extreme, it is a wise suggestion for mathematics.
There is structure to a theorem. Theorems have two parts, both of which we have already mentioned. These are the assumption and the conclusion (also called the antecedent and the consequent, respectively).
The assumption (or antecedent) of a theorem is a formal set of conditions that we accept as truth for the statement. A theorem's conclusion (or consequent) is the result that can be derived from the assumption according to a logical string of arguments.
This definition essentially states that the first part of a theorem is some assumption. This assumption is often considered the "if" statement. The second part of a theorem is what we think should happen as long as our assumption holds. This conclusion is often considered the "then" statement. To determine the assumption and conclusion of a theorem, it is recommended to rewrite the theorem in the form "If..., then...". When written in this form, the statement following the word "if" is the assumption, and the statement following the word "then" is the conclusion.
In the following theorem, what is the assumption, and what is the conclusion?
"If it is raining, then it must be cloudy."
- Answer
- The words following "if" are "it is raining." Therefore, the assumption is that it is raining. The words following "then" are "it must be cloudy." Hence, the conclusion is that it must be cloudy.
Just a single example is not enough to do justice to the idea of assumptions and conclusions. Let’s try another one.
In the following theorem, what is the assumption, and what is the conclusion?
"If I drink 14 gallons of water within 10 minutes, I will die."
- Answer
-
This is nearly written in "If..., then..." form. Rewriting it as such, we get, "If I drink 14 gallons of water within 10 minutes, then I will die." The assumption is that I drink 14 gallons of water in 10 minutes. The conclusion, based on that assumption, is that I will die.
The following example illustrates that critical thinking will be necessary to reword sentences or claims properly to see the assumptions and conclusions.
The following is a statement a mother made to her son.
"Eating ice cream causes polio."
What is the assumption, and what is the conclusion?
- Answer
-
In this statement (theorem), we are entirely missing the words "if" and "then." However, we can always reword it to contain these key words.
"If you eat ice cream, then you will get polio."
Now we can easily see that the assumption is that the son is eating ice cream, and the conclusion is that doing so will cause polio.
The claim in Example \( \PageIndex{ 4 } \) is not fictitious. A long-term study released in the late 1940s claimed this statement to be true. While some still cling to this belief, it is widely accepted that the interpretation of the data was flawed. First, polio is a communicable viral infection. Second, the high values of polio cases and ice cream consumption were from data collected during the summer. Of course, kids play in groups more during the warmer weather months, and warmer seasons will increase ice cream consumption. Finally, the low values of polio cases and ice cream consumption were from data collected during the winter. Again, this is pretty obvious – ice cream sales drop during colder months, and children tend to play less in groups when cold outside.
In other fields, terms like conjecture or hypothesis are used instead of theorem; however, each implies that the statement has not yet been proved and, therefore, the conclusion should be suspect. Only in mathematics do we have the pleasure of the theorem.
Since I am not speaking "math-speak" right now, the examples we have seen so far cannot officially be called theorems – they are technically conjectures or hypotheses. This is because someone, possibly you, could find a reason why the conjecture is false (remember, theorems have already been proved). Despite this, to keep our language less confusing, I will continue to use the word theorem to describe these and all future examples.
In each of the following theorems, state the assumption and the conclusion. Then, state a possible reason why the theorem could be false.
- If Nam eats ice cream cake, he will gain weight.
- Camille will get ticketed if she speeds down this street.
- Calculus II is a course that all engineering students at Cosumnes River College must take.
- Answer
-
- The assumption is that Nam is going to eat ice cream cake. The conclusion is the consequence of that action - he will gain weight. Is this theorem necessarily true? If Nam plans on exercising a lot after he eats the ice cream cake, then it could be a false statement.
- This is an example where rewriting as an "if..., then..." statement will help immensely. The statement says, "If Camille speeds down this street, then she will get ticketed." Thus, the assumption is that Camille is speeding down the street, and the conclusion is the consequence of that action, which is that she will be ticketed. Can you think of a way that this does not need to be true?
- The way this is written can throw some students off, but this is a theorem. The assumption is that we have an engineering student at Cosumnes River College. The conclusion is that she must take Calculus II ("If you are an engineering student at Cosumnes River College, then you must take Calculus II"). There is an easy way around this, though. If the student reads a Calculus textbook over the summer and tests out of Calculus II, she successfully avoids taking the course. So the theorem has a flaw.
As you can see from these examples, "theorems" outside of mathematics often have logical loopholes. Fortunately, true theorems (those within the field of mathematics) are generally flawless.
How important are theorems? You might say that without theorems, there is no thought. Without theorems, the world is just a bunch of statements with no conclusions. The idea of a theorem allows us to say,
"If it is raining, then it must be cloudy."
This statement makes sense, and almost serves as advice or warning. Without the theorem structure, either the conclusion or the assumption is gone, and we would only say,
"It is raining."
or
"It is cloudy."
These are just exclamations without any conclusion. It sounds more like we are stating a fact rather than suggesting what would happen "if" a condition was met.
As another example, consider the following "theorem."
"Camille will get ticketed if she speeds down this street."
As we stated in Example \( \PageIndex{ 5b } \), the assumption here is that Camille is speeding down the street, and the conclusion is that she will get ticketed. Removing the assumption leads to the odd statement,
"Camille will get ticketed."
Removing the conclusion leads to an even more nonsensical statement,
"Camille speeds down this street."
In either case, these are just statements without any assumptions or conclusions. They offer no advice, no warning, and no way out. With the assumption removed, Camille is getting ticketed – period! With the conclusion removed, Camille is definitely speeding down the street.
The following example illustrates a method to help determine whether or not a statement is a theorem. We essentially are trying to put it in the "if..., then..." form. While more complicated theorems are challenging to place into this form, the examples we see in this textbook are nice to work with and can be manipulated into this form somewhat easily.
Which of the following has the structure to be a theorem?
- Ron has the winning lottery ticket.
- When Linda goes to the store, she spends too much money.
- Who are you?
- Answer
-
- This cannot be written in the "if..., then..." form, so it is not a theorem. It would be a theorem if it were rewritten as, "If Ron has the winning lottery ticket, then he has at least a million dollars."
- This is a theorem because we can rewrite this as, "If Linda goes to the store, then she spends too much money."
- This is a far cry from a theorem. How can you fit an "if..., then..." statement here?
Proofs
The fact that a theorem (in mathematics) has been proved allows us to rely on its results without fear of exceptions. But what does it mean for a theorem to be proved? What is a proof?
A proof, in mathematics, is the validation of a proposition or theorem by the application of specified rules in a series of logical steps.
In layperson's terms, this means that a proof is a carefully constructed set of arguments laid out so that nobody, no matter how smart or creative, could refute the logic.
Let’s show how a proof is done by using a series of logical arguments to prove the following theorem.
If it is raining, then it must be cloudy.
- Proof
- Let us first assume that it is raining. Then, by the definition of rain (which we will assume we know), we have water falling from the sky. However, if water is falling from the sky, there must be moisture in the sky (since water is moisture). For there to be enough moisture in the sky to cause this moisture to coalesce into raindrops, there must be clouds. Therefore, if it is raining, it must be cloudy.
Okay, I am not a meteorologist, so I may have butchered that last bit of the proof; however, if I did start using more technical terms from meteorological science, I may bore you to death. I hope that example delivers the idea of a proof.
A subtle note about proofs is required here. The proof is necessary for a conjecture to be classified as a theorem; however, the proof is not considered part of the theorem. This means that proofs and theorems are different beasts. When someone in mathematics states a theorem, you have the right to request a proof of their statement; however, it is not necessary.
Every effort is made to prove all theorems stated in this textbook, and you should genuinely try to understand how each proof works; however, there may be times when a proof will be beyond the scope of our skills. For these theorems, I will mention that more advanced work is needed to derive the proof, or I may include their proofs in the appendix.
I have always thought of the theorem-proof relationship as a challenge. It’s as if someone walks up to you and says,
"If it is raining, then it must be cloudy. Prove it!"
You meet the challenger eye-to-eye and begin to explain why it has to be cloudy, given the fact that it is raining. Every statement you make is precise and clear. Each word is meant as another nail in the coffin of the challenger. Eventually, when all doubt as to the weakness of the conjecture has been erased, you claim victory. The challenger walks away, knowing that you are the wiser.
This is the essence of the proof.
Let’s try another proof to get the hang of it.
If you owned the car called the Aurora, you would own the only Aurora in the world.
- Proof
- Suppose that you owned the car called the Aurora. Since the designer only built one of these cars, you would own the only Aurora in the world.
Designed in 1957 by an eccentric New York priest as the world's safest car, the Aurora flopped, and only one was ever made. It had foam-filled bumpers, a roll cage, and seats that swiveled 180 degrees before a crash.
We could go for one more while we are at it. Prove the following theorem.
Every president with the last name of Adams finished his presidency in the 19th century.
- Proof
- Rewriting the theorem as an "if..., then..." statement, we have the equivalent statement, "If a president has the last name of Adams, then he finished his presidency in the 19th century."
The presidents with the last name of Adams are John Adams and John Quincy Adams. John Adams finished his presidency in 1801, and John Quincy Adams finished his presidency in 1829. Therefore, all presidents with the last name of Adams finished their presidency in the 19th century.
Before walking away from our discussion of theorems, I would like to mention that we encounter two subtypes of theorems in mathematics: the lemma and the corollary. At this point, the distinction between a theorem, a corollary, and a lemma is somewhat arbitrary. We will define these when we encounter them later in the textbook; however, you could start a new skill by looking in the index for the word lemma and reading its definition right now.