Download CHAPTER 0: WELCOME TO MATHEMATICS A Preface of Logic

CHAPTER 0: WELCOME TO MATHEMATICS
DAMIEN PITMAN
A Preface of Logic: What is Truth?
The first purpose of this course is to better understand how we verify truth
in mathematics. Outside of mathematics, truth is often verified by gathering
evidence, which suggests that something is true beyond reasonable doubt.
Mathematicians are so dubious about this process that we only deign to say
something is true once all doubt has been eliminated. We accomplish this is
by limiting the search for truth to a sort of game that has certain rules. The
following list begins to outline these rules.
(1) We assume true and false are understood without definition.
(2) We assume nothing can be true and false simultaneously.
(3) We define a statement or proposition to be any declarative sentence
that is either true or false, possibly depending upon what fixed values
any variables take. When we wish to emphasize that variables are not
yet fixed, we use the term open sentence.
(4) We agree that a proof is a valid deductive argument that verifies the truth
of a statement/proposition. We also agree that a valid deductive argument is based only on assumptions that we have accepted into the
mathematical system or statements/propositions that were previously
proved to be true.
(5) We define the truth value of a statement to be true if it was proved to
be true, or false if it was proved to be false. If the truth a statement has
not been proved, then it does not have a truth value and it is called a
conjecture. An argument that relies on a conjecture is not a proof, i.e.,
conjectures are not accepted into the mathematical system.
(6) An assumption is a declarative sentence that has not been proved and
so has no truth value, but is accepted as a basis for argument. We will
accept three types of assumptions into our mathematical system: definitions, undefined terms, and axioms. When we can stipulate the exact
meaning of a word, we write a declarative sentence that defines that
term. When we can not stipulate the meaning of a word to our satisfaction, but we need to use it, we accept that word as an undefined term
and make an attempt to explain away any possible confusion. When
we wish to specify an assumption we are willing to accept without
proof, we call that assumption an axiom.
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In these notes, we will use bold face text to indicate that the bolded term
is being defined and italics to emphasize text for other reasons. For example,
we will often use italics to emphasize terms that have not yet been defined.
These terms may be undefinable, defined later in the notes, or you may need
to look elsewhere for their definition.
We use the term theorem in reference to proven propositions that are especially important, say important enough to be accepted into the mathematical
canon and memorized. We use the term lemma for proven propositions that
are of interest primarily because they allow us to prove theorems. We use
corollary for propositions that require very little proof because they are easily
deduced from other known truths, such as theorems. Notice that it is impossible to mathematically define these terms because what is important or easy
is subjective.
When we define a term mathematically, we specify the conditions of the
definition and understand that these conditions must be met if we want to
apply the term and that the term always applies provided the conditions are
met. Thus, anyone playing the game of mathematics should agree on the
truth value of a statement asserting that an object satisfies a definition.
The fact that definitions and axioms are accepted without proof does not
mean that anything goes. A good definition should both clarify the meaning
of a term and be useful in proofs. A good axiom should be consistent with
reality as we know it, assume as little as possible, and yet form the foundation
upon which rich structures can be built. The assumption that nothing can be
true and false simultaneously is an axiom generally credited to Aristotle. It is
simultaneously obvious and essential, but can not be proved.
We wish to draw attention to one last assumption, which is often goes without mention. This is the assumption that specifies which rules of inference will
be allowed into the system. That is to say, we need to specify how we are
allowed to deduce truth values once we accept assumptions or proved propositions into the system. This topic will be discussed once we have defined
some more language in the chapter on logic.
To summarize, the game we call mathematics is an internally consistent
system within which we carefully state our assumptions and then deduce the
truth values of statements based on these assumptions and previously proved
statements. And now for some quotes meant to motivate you to question and
then improve your mathematical self.
: The fundamental cause of the trouble is that in the modern world the stupid
are cocksure while the intelligent are full of doubt. ∼ Bertrand Russel
: A mathematical theory is not to be considered complete until you have made it
so clear that you can explain it to the first man whom you meet on the street.
∼ David Hilbert
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Exercises
The lettered instructions are to be applied to each numbered item.
(a) Determine if what is written is a statement. If not, write your best
guess as to what true statement might have been intended.
(b) Conjecture the truth value for each statement.
(c) Carefully write a definition for each term or symbol below for which
the meaning is unclear.
(d) Based on your definitions, write arguments to support your conjectures.
(e) Compare your definitions and written arguments to those made by
your neighbors. If there are disagreements, try to resolve them.
(1) The number 2 is a factor of 6.
(2) The number 6 is a multiple of 2.
(3) The number 6 is divisible by 2.
(4) The number 2 is divisible by 6.
(5) The number 1/3 is a factor of the number 2.
(6) The number 2 is a factor of the number -6.
(7) The number -3 is a factor of the number -6.
(8) 2|6.
(9) 6|2.
(10) 2| − 6.
(11) 26 .
(12) 62 .
(13) The statement: a|b is the same as: There is some integer k satisfying
b = ka.
(14) Every integer is either even or odd and can not be both.
(15) Every integer is either divisible by 2 or by 3.
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Preface of Set Theory and Number Theory
• A set is a collection of objects that we call elements of the set.
• We write x ∈ S to indicate that x is an element of a set S.
• We write x ∈
/ S to indicate that x is not an element of a set S.
• We use braces {} to enclose the elements of a set and commas to separate elements inside the braces. When we list each element explicitly,
we are using roster notation. When we wish to indicate that a pattern is
to be followed and any object that fits the pattern is an element of the
set, we write . . ., in which case we are using ellipsis notation.
• The set of natural numbers, denoted N, is the set {1, 2, 3, . . .}.
• The set of whole numbers, denoted W, is the set {0, 1, 2, 3, . . .}.
• The set of integers, denoted Z, is the set {. . . , −2, −1, 0, 1, 2, 3, . . .}.
• The set of rational numbers, denoted Q, is the set of numbers that can
be expressed as the ratio of two integers.
• The set of real numbers, denoted R, is the set all numbers that are the
limit of a sequence of real numbers.
• The set of complex numbers, denoted C, is the set of numbers of the
form a + bi where a and b are real numbers and i denotes a nonreal
number that satisfies i2 = −1.
Theorem: (Division Algorithm) For any integer a and any positive integer b,
there are unique integers q and r, satisfying a = bq + r and 0 ≤ r < b.
Definition: In the division algorithm we refer to a as the dividend, b as the divisor,
q as the quotient, and r as the remainder.
Definition: To say an integer a is divisible by an integer b means the division
algorithm with a as the dividend and b as the divisor results in a
remainder of 0. When a is divisible by b we say that b is a divisor
or factor of a, or that b divides a, or that a is a multiple of b. The
statement: b divides a. is written in symbols as: b| a.
Definition: To say an integer is prime means it is greater than 1 and its only pos-
itive integer divisors are 1 and itself. To say an integer is composite
means it is greater than 1 and has a positive integer divisor other than
1 and itself.
Definition: To say an integer is even means it is divisible by 2 and to say it is odd
means its remainder is 1 when divided by 2.