Mathematical Proof/Print version
Table of Contents
1. Introduction
1. Logical Reasoning
2. Notation
2. Methods of Proof
1. Constructive Proof
2. Proof by Contrapositive
3. Proof by Contradiction
4. Proof by Induction
5. Counterexamples
3. Proof and Computers
Appendix
Answer Key
Symbols Used in this Book
Glossary
Introduction
If you've ever taken a math class or talked to a mathematician, you know that we don't talk like normal
people do--we have our own language that's 2-bit encoded. For example, when a waiter says "Would you
like corn or beans with your dinner?" you know that he wants the answer of "corn" or "beans" but not
"both." Well, logically, if you ask the question "Would you like corn or beans with your dinner?" the answer
could be "yes."
This chapter will acquaint you with logical statements, logical reasoning, and the cryptic language of
mathematicians.
Logical Reasoning
As discussed in the introduction, logical statements are different from common English. We will discuss
concepts like "or," "and," "if," "only if." (Here I would like to point out that in most mathematical papers it
is acceptable to use the term "we" when referring to oneself. This is considered polite by not commanding
the audience to do something nor excluding them from the discussion.)
Truth and Statements
A truth statement is one that is either true or false, not neither, and not both. Therefore, the sentence "This
sentence is false." is not a truth statement because its truth value cannot be determined. However, the
sentence "All people are cows." is a truth statement because its truth value can be determined, and is clearly
false, since there are some people that are not cows. In mathematics, normally this phrase is shortened to
statement to achieve conciseness and to avoid confusion.
The truth value of a statement is an evaluation of whether the statement is true (sometimes also referred or
abbreviated as 1 or T) or false (sometimes referred also as 0 or F). Thus, the truth value of "2 < 3" is true and
"4 > 5" is false.
In discussing logic and statements, it is common to use the letters P and Q as variables to denote statements.
Throughout this section, this convention will be followed. We will now discuss logical connectors, or
conjunctions that connect statements.
Logical Connectors
Or
The connector "or" is used to connect two statements and make a third statement whose truth value is based
on the first two. The statement "P or Q" is true if either P or Q is true, or if both of them are true. Therefore,
the statement, "The internet is complex or math is exciting." is true, given that either or both statements are
true. Another example, the statement "7 > 9 or 2 < 3" is true, because one of the connected statements is true.
The only time the statement "P or Q" is false is when both P and Q are false; an example of a statement
where both P and Q are false is the following "All people are American or all Americans are tall". Clearly,
the statement "All people are American" is false, and is false also the statement "all Americans are tall".
"Or" is sometimes denoted with the symbol . So
to as disjunction.
means "P or Q." This is also sometimes referred
And
"And" is used to connect two statements, just as "or"; however, "P and Q" is only true when both P and Q
are true, and false otherwise. The statement "The author of this article likes cheesecake and broccoli." is true
since the statements "The author of this article likes cheesecake." and "The author of this article likes
broccoli." are both true. Of course, it doesn't mean that I like them together on the same plate or in my mouth
at the same time as one might infer.
Another example might be the statement "3<6 and 4<6", which is true, since both statements "3<6" and
"4<6" are true. The statement "3<6 and 4>6" is not true since "4>6" is false.
"And" is sometimes denoted as .
means "P and Q." This is referred to as conjunction.
Implications and Conditional Statements
This is another class of logical connectors, like "and" and "or." The three implication types are "if" , "only
if" , and "if and only if" or "iff"
. See also Necessary and sufficient conditions. The vast majority of
mathematical proofs deal with statements such as
(P only if Q) or
(P if and only if Q).
Necessary
To say that "Q is necessary for P" means that P cannot exist without Q. Another way of saying this is "P
only if Q" or "if P then Q". The notation for this is
(P implies Q).
The statement "Sunday is necessary for Easter" is true because if it is Easter, then it must be also Sunday.
Yet, the statement "Water is necessary for humans" is, in a logical sense, false because something that is a
human is probably not water as well. (However, the statement "Having water is necessary for humans" is
true, because to be a human being something must contain water.) The statement "P only if Q" is false only
when P is true and Q is false, because Q is a necessary condition (it must be true) for P to be true.
Sufficient
To say that "P is sufficient for Q" means "P cannot exist without Q" or "if P then Q". The notation for this is
. This statement is only false when P is true and Q is false. This is the same statement as
,
but the roles are reversed; the former, however, is more common.
The interesting thing about an implication (an if-then statement) is that in the case that P is false,
is always true. Thus, "If the world is flat, then the Sun does not exist" is a true statement. This kind of a
statement is called vacuously true because Q can be replaced with any statement and the implication is still
true. In fact, the mathematician Bertrand Russell boasted, "Give me any false statement and any other
statement to prove and I will prove it". Given the statement "1 = 2" he was asked to prove that he was God.
"Consider the set
", he replied, "since 1 = 2, the two elements are one, and therefore
Russell = God".
"If" and "only if" statements are called conditional statements. These two are related, in fact they are the
converse (https://en.wikipedia.org/?title=Converse_(logic)) of each other. Converse simply means to switch
the direction of the arrow. (The converse of
is
.)
Necessary and Sufficient
To say that "P is necessary and sufficient for Q" means "P is true exactly when Q is true" or "P implies Q
and Q implies P". This means that P and Q always have the same truth value—that is, they are both true or
both false simultaneously. In mathematical symbols this is expressed as
. This is referred to as a
biconditional. When
is a true statement, we say that P and Q are logically equivalent.
Modifiers
We have explored truth statements and logical connectors used to create new statements. We will now look
at two more variations of logical statements.
Negation
Negation refers to toggling a truth statement's truth value. The result is not always negative, or false, it is
only the opposite of the original. Thus, the negation of "false" is "true." This is denoted by a tilde (~) or the
negation symbol .
It will be shown below in a truth table that the negation of the statement
, which is
, is
equivalent to
. Thus, to use loose terminology, "not or" is "and" (similarly, "not and" is "or").
Contrapositive
The contrapositive is related to negation. However, it is only defined for a conditional statement. It basically
means negate both P and Q and switch the direction of the conditional. That is, the contrapositive of
is
; or, equivalently,
. It will be shown below that the contrapositive of
a statement is equivalent to the statement itself.
Truth Tables
A truth table is helpful in visualizing all of these logical relations. The table is filled in by considering all
possible combinations of true and false for P and Q and then filling in the results for the various connectors
mentioned above.
The Or and And Table
OR and AND
P
Q
T
T
F
F
T
F
T
F
T
T
T
F
T
F
F
F
If, Only if, and If and only if Table
If, only if, and iff
P
Q
T
T
F
F
T
F
T
F
T
F
T
T
T
T
F
T
T
F
F
T
Note that the
column can be obtained from
useful in helping beginners understand logical relationships.
. Truth tables are very
Negation and Contrapositive Table
Negation and Contrapositive
P
Q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
T
F
T
T
Exercises
In the following exercises, P, Q, R, and S will represent truth statements.
A. Construct the truth tables for the following statements, and give their converse and contrapositive:
1.
1.
2.
3.
4.
5.
B. Negate the following statements:
1.
1.
2.
3.
Notation
While a comprehensive list of notation is included in the appendix, that is meant mostly as a reference tool
to refresh the reader of what notation means. This section is to introduce the notation to the reader and
explain its usage.
Basic Set Theory
This is not meant to be a comprehensive or rigorous definition of set theory. We will define a minimal
amount of set-theoretical objects, so that the concept of mathematical thinking can be understood. In this
book, we will use capital letters for sets and lowercase letters for elements of sets. This convention is not to
discourage creativity or to bore your socks off, but to avoid confusion.
Axioms
An axiom is something that is assumed, or believed to be true. It is where mathematical proof starts; you
cannot prove the axioms, you merely believe them and use them to prove other things. There are different
sets of axioms, the most current and widely-used being Zermelo–Fraenkel set theory. I will give a list of
axioms here that will suffice for the studies in this book.
1. A set exists. (A set, for our purposes, will be a collection of objects that we will call elements. A set is
said to contain its elements, and the elements are said to be contained in the set.)
2. An empty set exists. This set will be denoted as and will contain no elements.
3. Two sets are equal if and only if they contain the same elements.
4. If and are sets, then there is a set containing only the elements of and . This is called the
union of and .
5. If is a set and
is a truth statement defined for all contained in , then there is a set so
that is in whenever
is true.
6. The set of counting numbers
exists; or, an infinite set exists.
Some of these are worded rather formally, which is a tendency for mathematicians. First of all, let's explain
why we need these axioms.
The first axiom says "a set exists." So you might ask—don't we know that it exists? Can't we just define it to
exist? The answer is, yes, and that's why it's an axiom. Axioms are supposed to be self-evident truths. Now
that we've established the fact that sets exist, why would we want one with no elements? Well, the empty set
turns out to be both a very useful and a very annoying set. You'll learn to be good friends with the empty set
by the end of the text.
Axiom 3 could be considered a definition, rather than an axiom, of what we mean when we say that two sets
are equal. Axiom 4 just says that if we have two sets then we can get a new set with all of those elements in
it. For example, the set of all people and all dogs.
The fifth axiom is probably the most confusing. All it says is that if we have a set and we want to pick out
certain elements, we can do that. For example, out of the set of all integers, we can choose the ones that are
even, or the ones that are positive, or the ones that are perfect squares.
Finally, the infinity axiom is nice because we will do lots of things with infinite sets.
Definitions
As mentioned above, a set will be a collection of elements. For example, let be the set of all cheesecakes,
and let be the set of all things that are chocolate. Mathematically, this would be denoted as:
The vertical bar | is read "such that". We can select elements in that way by using Axiom 5 from above. In
the case of , the predicate (truth statement)
used to select elements is:
Note that we have implicitly assumed the existence of a universal set of all elements over which we are
making a selection. In the example above, this universal set could be the set of all pastries. In general, if a
universal set is not specified, we shall assume that we are talking about the real numbers . Thus,
can be read as: " is the set of all real numbers strictly greater than ."
To say that is an element of is equivalent to saying that
mathematically by
(x belongs to the set A) and
(x does not belong to the set A).
contains . These concepts are notated
. If is not a member of then we write
The union of and is defined in axiom 4. It consists of all the elements in
. We can also use the | to say
The intersection of
is
.
and
is the set containing all things that are in both
and
and
and is denoted
. The notation for this
If
then and are said to be disjoint. This means that there is nothing that is in both sets.
For example, if is the set of all even integers and is the set of all odd integers, then they are disjoint.
Notice that the logical connectors
coincide with the set operators
. This is deliberate, since the
concepts are related. This is obvious when the two symbols are juxtaposed:
Quantifiers
Quantifiers are used to establish what elements are currently being discussed. They are like adjectives in
English—they tell how much or what kind of thing we're talking about.
For All
The most common quantifier is for all. This is written mathematically as . It is also "for each" or "for
every." It is used to make statements like "All humans have eyeballs." That is, if
is the set of all humans
and is the set of all things with eyeballs, then
which is read "For all in , is in ." This introduces a special relationship between sets that is called a
subset. In this case,
is a subset of since every element of
is also in . (For the sake of the logical
argument, just assume there aren't any people without eyeballs.) We write this as
This notation,
, is ambiguous because some authors use it to mean just a subset, while others use it
to mean a proper subset (meaning there is an element of that is not in , and thus and are not
equal) and use
to denote that is a subset of . In this book we will go with the convention that
could mean that is a proper subset of or that = , and we will use
when we wish
to emphasize that
.
There Exists
This is probably as common as for all and is equally useful. Its mathematical symbol is . It is almost always
followed by a "such that" statement. For example, "There exists a computer that has 8GB of RAM." and
are often used in pairs, such as, "Everyone has a mother.", or, worded logically, "For each human, there
exists a mother for that human.". Let H be the set of all humans and M be the set of all mothers. Then we
have
or, when
and
are understood,
This quantifier is also read as there is or there are. To signify that there is only one of something, we say
"There exists a unique..." and place an exclamation point after the exists symbol: .
In the same way that "not and" gives "or", "not for all" gives "there exists." That is, the opposite of the
statement "All cheesecakes are chocolate." is "There is a cheesecake that is not chocolate." In logical terms,
The above statement is read "the negation of 'For all x in A, P(x) is true.' is 'There is an x in A such that P(x)
is false."
Such That
As we have seen, such that can be used in at least two cases: in conjunction with there exists and in picking
out elements of a set. Of course, if you think about it, these two are really the same application because the
statement "there exists" gives you the set of all things that exist, and the such that statement decreases the
size of that set to focus just on the things in which you are interested. Such that is normally denoted as a
colon(:) or as a vertical bar (|), and sometimes as "s.t."
Without Loss of Generality (https://en.wikipedia.org/wiki/Without_loss_of_generality)
This is a very helpful phrase in making proofs more concise and less redundant. For example, assume we
have two integers x and y and that we know one of them is odd and one of them is even. Instead of trying to
do two different parallel proofs, one where we assume that x is even and y is odd and another where we
assume that y is even and x is odd, we simply say "Without loss of generality, assume x is even." Then we
continue with the proof. This is done since the exact same argument applies if y actually was the even
number, all we have to do is relabel x and y.
The Universe
(https://en.wikipedia.org/wiki/Universe_(mathematics))
Now, we're not going to begin a discussion of astronomy. In mathematics, the universe is overall, biggest set
that your discussion is referring to. For example, if the universe is not restricted, then the set of all things
would truly be the set containing every single thing. However, if your universe is the set of all things on
Earth, then the "set of all things" would not include Jupiter, since Jupiter is not on Earth.
Difference
In arithmetic, difference means the distance between two numbers—how far apart they are on the number
line. In set theory, difference means something slightly different, but the same notation is used. (
denotes the difference of A and B.) The difference is the set of all things that are in A that are not in B.
In normal English, we use this concept when we say things like "Everyone with no demerits gets an A in this
course."
Complement
The complement of a set contains everything that is not in the
original set. This definition only makes sense when a universe is
understood. The complement is usually denoted by
. If U is the
universe, then the complement of A is defined to be
.
The diagram to the right is a Venn diagram. A Venn diagram shows
the relationships of sets. Note that in the figure, U is the universe,
and A and B are sets in U. The blue portion is the complement of
. This is a general drawing, since it is not known whether
there are elements in A and B. If
is known to be empty, then
they may be drawn disjoint.
A Venn Diagram of A, B, and U.
The complement of a set is essentially the same as the negation of a statement. That is,
if
, then
.
Thus, complements are used when saying what something is not.
Exercises
1. Express the following statements in terms of sets, using difference or complement.
1. All people that have two legs.
2. All mythological creatures that are not Greek.
3. All pudding pies that have no cream on top.
2. Draw a Venn Diagram to illustrate the following.
1.
2.
3.
4.
5.
3. Negate the following statements.
1.
2. All quick, brown foxes jump over some lazy dog.
Methods of Proof
There are many different ways to prove things in mathematics. This chapter will introduce some of those
methods.
Constructive Proof
A constructive proof is the most basic kind of proof there is. It is a proof that starts with a hypothesis, and a
person uses a series of logical steps and a list of axioms, to arrive at a conclusion.
Parts of a theorem
A theorem (https://en.wikipedia.org/wiki/Theorem) is a proven statement that was constructed using
previously proven statements, such as theorems, or constructed using axioms. Some theorems are very
complicated and involved, so we will discuss their different parts.
Hypothesis
The hypothesis is the "if" statement of a theorem. In a way, it is similar to an axiom because it is assumed to
be true in order to prove a theorem. We will consider a simple example.
Theorem 2.1.1. If A and B are sets such that
and
, then A=B.
In this theorem, the hypothesis is everything before the word "then." This is a very simple proof. We need to
prove that for every x,
. For the purpose of analyzing proofs, we will define
and
. The most common way to prove an "if and only if" statement is to
prove necessity and sufficiency separately
So we start by showing that
Therefore, we assume that P(x) is true. That is,
Since we assumed, by hypothesis, that
, we know that
, which means that
Q(x) is true.
Now we show that
we know that
, so we assume that Q(x) is true. This means that
we know that
so P(x) is true.
By these two conclusions, we see that
. Since
Now, by axiom 3, A=B, since
This concludes the proof. This is a very trivial proof, but
its point was to show how to use a hypothesis or set of hypotheses in order to reach the desired conclusion.
This method here is the most common in proving that two sets are equal. You prove that each set is a subset
of the other.
Conclusion
The part of the theorem after the word "then" is called the conclusion. The proof of a theorem is merely the
logical connection between the hypothesis and the conclusion. Once you've seen and proved a few theorems,
a conclusion is almost predictable. For example, what conclusion would you naturally draw from the
following two statements?
1. All Americans are people.
2. All people live on Earth.
These two statements are the hypotheses. To word this as a theorem, we would have "If all Americans are
people and all people live on Earth, then all Americans live on Earth." This statement is what most people
would call completely obvious and requires no proof. However, to show how this concept is applied in
mathematics, we will abstract this theorem and prove it.
Theorem 2.1.2. If
and
, then
.
To see how this relates to our problem, let A be the set of all Americans, B the set of all people, and C the set
of all things that live on Earth.
To show that
, we need to show that
so
Also by hypothesis,
we have shown that
, so
So we suppose
By hypothesis,
Since this was true for any arbitrary
Definition
While a definition is not usually part of a theorem, they are commonly introduced immediately before a
theorem, in order to make the theorem make sense or to help in proving it.
Definition 2.1.3. If a set A has only finitely many elements, then the order of A, denoted by |A|, is the
number of elements in A.
This definition gives meaning to the following theorem.
Theorem 2.1.4. If A and B are finite sets such that A = B, then |A|=|B|.
Here we take advantage of the fact that A is a finite set. Let n be the integer such that |A| = n. Then index the
elements of A so that
Now
, we have
. So we see
that B has at least n elements, that is
Also, every element of B is in A (by hypothesis), so it
follows that there are no more elements in B than there are in A, so
, thus |B| = n = |A|, which
concludes the proof.
Pseudo Theorems
There are different terms that mathematicians like to use in stating mathematical results. Theorem is
probably the most common and well-known, especially to non-mathematicians. There are, however, a couple
other main terms used in the mathematical world. For lack of a better term, I have lumped them together in
the category of "pseudo-theorems", since they are like theorems, but are different.
Lemma
A lemma (https://en.wikipedia.org/wiki/Lemma_(mathematics)) is a "small theorem." When a result is less
profound, more trivial, or boring, it can be called a lemma. A lemma is also used to make the proof of a
theorem shorter. That is, if a chunk of a proof can be pulled off and proved separately, then it is called a
lemma and the proof of the theorem will say something to the effect of "as proved in the lemma."
For example, the following lemma will help to make the proof of Theorem 2.1.4
(https://en.wikibooks.org/w/index.php?
title=Mathematical_Proof/Methods_of_Proof/Constructive_Proof&stable=0#Definition) more concise.
Lemma 2.1.5. If A and B are finite sets and
then
As you might guess, this is one motivation for the use of the symbol
.
, since it is similar in appearance to <.
Let n = |A|. Then number the elements of A, so
Then for each i from 1 to n we
see that
, which means that B has at least n different elements, or that
which is
what we were trying to prove.
Now if we use this lemma twice on Theorem 2.1.4, we will get a very brief proof. Since
we know
that
Also, since
, we see that
Now we use a fact about numbers, that if
and
, it must follow that x = y.
Corollary
A corollary (https://en.wikipedia.org/wiki/Corollary) is similar to a lemma in that it is usually a small and
not as important as a theorem. However, a corollary is usually a result that follows almost immediately from
a theorem.
Assume that we had proved the theorem that "All people are pigs." Then a corollary would be "People who
have two legs are pigs." which clearly follows from the first result. A slightly more interesting corollary
would be "People can be sold for bacon when they die." since it is common knowledge that bacon comes
from pigs.
So we see that a corollary is something that follows from a preceding theorem with minimal argument to
support it. It is either "common sense" or "obvious" to the reader that it is a direct consequence of the
theorem.
Exercises
1. Prove that the following sets are equal. Verify it with a truth table or a Venn Diagram. You may
assume that A, B, and C are nonempty sets. Also assume that U is the universe.
1.
2.
3.
4.
5.
2. Prove that if A and B are finite sets then
and that equality holds when
Something to think about
We have defined the order or size of a set for a finite set. Would it make sense to define this order for
an infinite set? How would you tell whether two infinite sets are the same size?
If you know that
can you show that
Proof by Contrapositive
The contrapositive of a statement negates the conclusion as well as the hypothesis. It is logically equivalent
to the original statement asserted. Often it is easier to prove the contrapositive than the original statement.
This section will demonstrate this fact.
The Concept
The idea of the contrapositive is proving the statement "There is no x such that P(x) is false." as opposed to
"P(x) is true for every x."
This is not to be confused with a Proof by Contradiction. If we are trying to prove the statement
,
we can do it constructively, by assuming that P is true and showing that the logical conclusion is that Q is
also true. The contrapositive of this statement is
, so we assume that Q is false and show that
the logical conclusion is that P is also false. However, in a proof by contradiction, we assume that P is true
and Q is false and arrive at some sort of illogical statement such as "1=2."
A proof revisited
The most basic example would be to redo a proof given in the last section. We proved Theorem 2.1.4 to be
true by the constructive method. Now we can prove the same result using the contrapositive method. For
ease of reading, I will change the wording of the theorem.
Theorem 2.2.1 (also Theorem 2.1.4). If A and B are finite sets and
, then
.
The wording is probably more natural in the 2.1.4 version, but this shows how the proof is to be done. We
assume that we have two finite sets A and B and that they do not have the same number of elements. So let
and
. Then, number the elements in A and B, so
and
. Since
, either
or
. Without loss of generality, we
assume that
. Consider the set
. Since A has only n elements, we can take out at most n
elements from B, leaving at least m-n elements in B-A. This shows that there is at least one element in B that
is not in A, therefore
.
Arithmetic
I'm sure we all know how to do arithmetic already, but mathematicians like to be "rigorous." That means that
we like to have clear definitions of everything we use.
Axiom 7. The numbers 0 and 1 exist.
Definition 2.2.2. The operator + is defined so that 1+0=1 and
is an infinite set. We write the elements of this set as
and define
.
Definition 2.2.3. The operator is defined so that
We also define
and
The above definitions are just formalities. We know what numbers are and how they work. This is just a
mathematical definition. Notice that only integer multiplication has been defined. Now we need one more
definition to prove the following theorem.
Definition 2.2.4. An integer is said to be even if it is a multiple of two. That is, 'n' is even if n = 2k for
some integer k. If this is not true, then it is said to be odd. The property of whether a number is even or
odd is its parity.
Theorem 2.2.5. If x and y are integers such that
is odd, then
To prove this by contrapositive, we assume that
, which by Definition 2.2.3 is
and show that
is even. If
, then
, so
is a multiple of 2 and is therefore even.
Biconditionals
Proofs by contrapositive are very helpful in proving biconditional statements. Recall that a biconditional is
of the form
(P if and only if Q). To prove a biconditional we need to prove that
and
However, if we use the contrapositive, we can show
and
More Arithmetic
Prime numbers are very interesting in number theory, but also in arithmetic. In fact, the Fundamental
Theorem of Arithmetic has to do with primes. Here we will not give a proof, just a statement of the the
theorem.
Definition 2.2.6. A prime number is a positive integer that has no multiple other than 1 and itself. A
number that is not prime is called composite.
Theorem 2.2.7 (Fundamental Theorem of Arithmetic). Every integer has a unique factorization of
primes, excluding reorderings.
This theorem will be very useful in the proof of the next theorem.
Theorem 2.2.8. An integer n is even if and only if its square
Proof
is even.
First we do a constructive proof. Suppose that n is an even integer. Then by defintion of
even,
for some integer k. Then
.
Since
is an integer because it is the product of integers, we see that
is even.
This shows the "only-if" part of the theorem.
To show the "if" part, we use a proof by contrapositive. Assume that n is not even, or that
n is odd. Let
be the prime factorization of n. Then none of the
are 2 for any i. We consider the square
multiples that are equal to 2, so we conclude that
and notice that there are no
is odd. This proves the theorem.
Exercises
1. Prove the following by contrapositive:
1. An integer n is even if and only if n+1 is odd.
2. If n and m have the same parity then n+m is even.
Proof by Contradiction
The method of proof by contradiction is to assume that a statement is not true and then to show that that
assumption leads to a contradiction. In the case of trying to prove
this is equivalent to assuming
that
That is, to assume that is true and is false. This is known by its latin reductio ad
absurdum (reduction to absurdity), since it ends with a statement that cannot be true.
Square root of 2
A good example of this is by proving that
is irrational. Proving this directly (via constructive proof)
would probably be very difficult (if not impossible). However, by contradiction we have a fairly simple
proof.
Proposition 2.3.1.
Proof: Assume
is rational. Then
no common factors).[1] and
But since
is even,
and
are relatively prime integers (a and b have
. So
must be even as well, since the square of an odd number is also odd. Then we have
, or
so
, where
.
The same argument can now be applied to to find
. However, this contradicts the original
assumption that a and b are relatively prime, and the above is impossible. Therefore, we must conclude that
is irrational.
Of course, we now note that there was nothing in this proof that was special about 2, except the fact that it
was prime. That's what allowed us to say that was even since we knew that
was even. Note that this
would not work for 4 (mainly because
) because
does not imply that
More Thoughts
In English, the procedure is this: Assert that a statement is false, and then prove yourself wrong. (Thereby
proving the original statement was true.) This is a form of Modus Tollens.
For many students, the method of proof by contradiction is a tremendous gift and a trojan horse, both of
which follow from how strong the method is. In fact, the apt reader might have already noticed that both the
constructive method and contrapositive method can be derived from that of contradiction.
Assume
Prove
Contradiction
However, its reach goes farther than even that, since the contradiction can be anything. Even if we ignore the
criticisms from constuctivism, this broad scope hides what you lose; namely, you lose well-defined direction
and conclusion, both of which have to be replaced with intuition.
Lastly, even in nonconstructive company, using the method in the first row of the table above is considered
bad form (that is, proving something by pseudo-constructive proof), since the proof-by-contradiction part of
it is nothing more than excess baggage.
1. It is a simple exercise to see that any rational number may be written in this form.
Proof by Induction
The beauty of induction is that it allows a theorem to be proven true where an infinite number of cases exist
without exploring each case individually. Induction is analogous to an infinite row of dominoes with each
domino standing on its end. If you want to make all the dominoes fall, you can either:
1. push on the first one, wait to see what happens, and then check each domino afterwards (which may
take a long time if there's an infinite number of dominoes!)
2. or you can prove that if any domino falls, then it will cause the domino after it to fall. (i.e. if the first
one falls then the second one will fall, and if the second one falls then the third one will fall, etc.)
Induction, essentially, is the methodology outlined in point 2.
Parts of Induction
Induction is composed of three parts:
1. The Base Case (in the domino analogy, this shows the first domino will fall)
2. The Induction Hypothesis (in the domino analogy, we assume that a particular domino will fall)
3. The Inductive Step (in the domino analogy, we prove that the domino we assume will fall will cause
the next domino to fall)
Weak Induction
Weak induction is used to show that a given property holds for all members of a countable inductive set, this
usually is used for the set of natural numbers.
Weak induction for proving a statement
(that depends on
) relies on two steps:
is true for a certain base step. Usually the base case is
. That is, given that
or
is true,
is also true.
If these two properties hold, one may induce that the property holds for all elements in the set in question.
Returning to the example, if you are sure that you called your neighbor, and you knew that everyone who
was called in turn called his/her neighbor, then you would be guaranteed that everyone on the block had
been called (assuming you had a linear block, or that it curved around nicely).
Examples
The first example of a proof by induction is always 'the sum of the first n terms:'
Theorem 2.4.1. For any fixed
Proof:
Base step:
, therefore the base case holds.
Inductive step: Assume that
. Consider
.
So the inductive case holds. Now by induction we see that the theorem is true.
Reverse Induction
Reverse induction is a seldom-used method of using an inductive step that uses a negative in the inductive
step. It is a minor variant of weak induction. The process still applies only to countable sets, generally the set
of whole numbers or integers, and will frequently stop at 1 or 0, rather than working for all positive
numbers.
Reverse induction works in the following case.
The property holds for a given value, say
.
Given that the property holds for a given case, say
.
Then the property holds for all values
, Show that the property holds for
.
I learnt today from http://people.math.carleton.ca/~ckfong/hs14a.pdf (see example there) that reverse
indcution is also usable in the general case: "to establish the validity of a sequence of propositions Pn (n ≥
1), it is enough to establish the following
(a) Pn is valid for infinitely many n.
(b) If Pn+1 is valid, then so is Pn.
It can be the case that we can easily prove P1 and if P for n=m so P for n=2m. In this case we have (a) for
the infinitely many n = 2 exp k (for k >= 0).
Strong Induction
In weak induction, for the inductive step, we only required that for a given , its immediate predecessor (
) satisfies the theorem (i.e.,
is true). In strong induction, we require that not only the
immediate predecessor, but all predecessors of satisfy the theorem. The variation in the inductive step is:
If
is true for all
then
is true.
The reason this is called strong induction is fairly obvious--the hypothesis in the inductive step is much
stronger than the hypothesis is in the case of weak induction. Of course, for finite induction it turns out to be
the same hypothesis, but in the case of transfinite sets, weak induction is not even well-defined, since some
sets have elements that do not have an immediate predecessor.
Transfinite Induction
Used in proving theorems involving transfinite cardinals. This technique is used in set theory to prove
properties of cardinals, since there is rarely another way to go about it.
Inductive Set
We first define the notion of a well-ordered set. A set
is well-ordered if there is a total order < on
that whenever
is non-empty, there is a least-element in . That is,
such that
.
An inductive set is a set
1.
2. If
(where
then
and
such that the following hold:
is the least element of
such that
)
Of course, you look at that and say "Wait a minute. That means that
!" And, of course you'd be
right. That's exactly why induction works. The principle of induction is the theorem that says:
Theorem 2.4.2. If
.
is a non-empty well-ordered set and
is an inductive subset of
then
The proof of this theorem is left as a very simple exercise. Here we note that the set of natural numbers is
clearly well-ordered with the normal order that you are familiar with, so is an inductive set. If you accept
the axiom of choice, then it follows that every set can be well-ordered.
Counterexamples
A proof by counterexample is not technically a proof. It is merely a way of showing that a given statement
cannot possibly be correct by showing an instance that contradicts a universal statement. For example, if you
are trying to prove the statement "All cheesecakes are baked in Alaska." and you did not know whether to
prove it by contrapositive or contradiction, all I would have to do is bake a cheesecake right in front of you
here in Texas and then you would know that your efforts had been in vain.
Consider the following statement:
Every set is countable.
Of course a counterexample to this would be the interval from
countable, we assume that it is, so then there is a bijective function
write the elements of
in a list here we write the numbers in
. To show that this interval isn't
since is bijective we may
in their decimal expansions.
For the purposes of the proof, we need each real number to be uniquely defined by a decimal. This is of
course true in almost all cases (take .01, for instance... it's equal to .00999999999...). In fact, the cases when
it's not true are exactly the cases when the decimal ends in a string of zeros, or "terminates" (a separate proof
would be needed for this, but for now accept it as true). Therefore, let the decimals all be in "nonterminating
form," so that there's another bijection between the decimal expansions and the real numbers.
Now let
if
is odd and 1 if
is even. Now I claim that
here. To show this suppose it were then for some
if
is odd and
if
, this is not possible so we conclude that
not in the list, this shows we may not have a surjection from
is not listed
but
is
Although this is a counterexample, we still had to PROVE that it was in fact a counterexample and in doing
so used both a proof by contradiction (this was the overall method of the proof) by a construction (of
). Although there may be more then one counterexample to any given false claim, you
must always provide by a proof or argument that your counterexample works.
Computer Proofs
== The Curry-Howard Correspondance ==
The Curry-Howard correspondence establish a close relationship between computer programs and
mathematical proofs.
Appendix
Answer Key
Logical Reasoning
1.1, Logical Reasoning.
These are the answers to the exercises in section 1.1, Logical Reasoning
1. Truth tables
P
1.
Q
T
T
F
T
F
T
F
F
T
T
T
T
F
F
P
2.
Q
T
T
F
F
T
F
T
F
T
F
F
T
F
T
F
T
T
T
3.
P
Q
R
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
T
T
F
F
T
F
T
F
T
F
T
T
4.
P
Q
R
S
T
T
T
T
T
T
T
T
F
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
T
F
T
F
T
F
T
F
T
F
T
T
T
T
T
F
F
F
F
F
F
F
F
F
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
F
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
T
F
F
F
T
F
F
F
F
T
T
F
T
T
T
5.
P
Q
R
S
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
T
T
T
F
F
F
F
T
T
T
T
T
T
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
T
T
F
T
F
F
T
T
T
T
T
T
T
T
2. Negated statements
1.
2.
3.
Notation
These are the answers to the exercises in section 1.2, Notation
1. Use set theory notation
1.
, where P is the set of all people and L is the set of all things with two legs.
2.
, where M is the set of all mythological creatures and G is the set of all things that are
Greek.
3.
, where A is the set of all pies, B is the set of all things with pudding in them, and
C is the set of all things with cream on the top.
2. Venn Diagrams
3. Negated statements
1.
2. There is a quick, brown fox that does not jump over any lazy dog.
Symbols Used in this Book
This is a list of all the mathematical symbols used in this book.
Closed interval notation. Signifies the set of all numbers between a and b (a and b included)
A logical "or" connector. A truth statement whose truth value is true if either of the two given
statements is true and false if they are both false.
A logical "and" connector. A truth statement whose truth value is true only if both of the two
given statements is true and false otherwise.
A logical "not" unary operator. A truth statement whose value is opposite of the given
statement.
{}
Set delimiters. A set may be defined explicitly (e.g.
), or pseudo-explicitly
by giving a pattern (e.g.
. It may also be defined with a given rule (e.g.
, the set of all x such that P(x) is true).
The "element of" binary operator. This shows element inclusion in a set. If x is an element of A
we write
The "set inclusion" or "subset" binary operator. If all the elements of A are in B, then we say that
A is a subset of B and write
Note that in this book,
when
The union of two sets. A set containing all elements of two given sets.
The intersection of two sets. A set containing all the elements that are in both of two given sets.
Glossary
This glossary is mostly just for a quick reminder of terms learned in the book and is not meant to be
comprehensive or rigorous. Please visit Wikipedia or Wiktionary for more detail.
A
Back to top
Arithmetic The science of addition and multiplication (subtraction and division are included, since they are the
inverse operations of addition and multiplication). Proof by Contrapositive
Axiom A self-evident truth. It is the foundation of logical reasoning. A statement that is accepted as true
without proof, which may be assumed in proving that other things are true.Notation
B
Back to top
Basis A collection of open sets in a set
set
Proof by Contradiction
such that the intersection of any two open sets in
contains a
C
Back to top
Closed set The complement of an open set in a topological space. Proof by Contradiction
Conclusion The result of a given conditional statement. (The "then" clause of a theorem.) This is also sometimes
referred to as the result. Constructive Proof
Conditional statement
An "if" or an "only-if" statement. It is conditional because its truth value is determined by the truth
value of two other statements. Logical Reasoning
Contrapositive The converse and negation of a conditional. The contrapositive of
Reasoning
Converse The "reverse" of a conditional statement. The converse of
is
is
. Logical
. Logical Reasoning
Corollary That which follows, usually without any necessary argument, from a given result. Constructive Proof
D
Back to top
Divisor See factor.
Divide An integer n divides an integer m, if n is a factor of m, equivalently, if m is a multiple of n, or,
equivalently, if there's a integer k such that
. Proof by Contrapositive
E
Back to top
Element One of the objects in a set. Notation
Equivalent See Logically Equivalent.
F
Back to top
Factor An integer that divides a given integer. (e.g. 3 is a factor of 6.) This is the "opposite" of multiple.
Proof by Contrapositive
L
Back to top
Lemma A result whose proof is fairly simple or one that is used to simplify or break down a larger argument.
Constructive Proof
Logcially Equivalent
Two statements that are simultaneously true or simultaneously false are logically equivalent. Logical
Reasoning
M
Back to top
Multiple An integer obtained by multiplying two integers together. (e.g. 4 is a mulitple of 2). This is the
"opposite" of factor. Proof by Contrapositive
N
Back to top
Negation The opposite of a truth statement. The negation of trueis falseand vice-versa. Logical Reasoning
O
Back to top
Open set A set that is an element of a topology
defined on a set
Proof by Contradiction
R
Back to top
Result A lemma, theorem, or corollary. A statement of "if-then" that has been proven to be true. Also, the
conclusion of such a statement. Constructive Proof
S
Back to top
Set A collection of items, or elements. Notation
Statement See Truth Statement.
T
Back to top
Theorem A main result. Usually the proof is somewhat involved and the result is interesting and useful.
Constructive Proof
Topological Space A set
together with a topology
that satisfy the topology axioms. Proof by Contradiction
Topology A collection of subsets of a given set that satisfy the topology axioms. Proof by Contradiction
Truth Statement
A statement whose truth value can be determined. Therefore, it is either true or false. Logical
Reasoning
Truth Value
The assessment of whether a statement is true or false. Logical Reasoning
License
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or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
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