Proof: Introduction to Higher
Mathematics, Fifth Edition
by Warren W. Esty and Norah C. Esty
The content
includes the usual content of a “methods of proof
” or “transition to higher mathematics” course.
Mathematical logic is covered thoroughly (truth tables and
quantifiers), the basic forms of proofs are covered (direct proofs,
indirect proofs, proofs by mathematical induction, etc.), and there are
many exercises for students. The main difference between our text and
others with similar goals is probably that we take less for granted. We
do not assume that students are already fluent in the language of
mathematics and merely need minor guidance when it comes to
constructing proofs. On the contrary, we see many math and math
education majors who have very little background in logic. Many
did not do a lot of proofs when they took geometry.
Consequently,
our text spends more time early in the course on basic logic and
interpretation of symbolic sentences than most other texts.
Unexpected
Things
We
do That You
Will Like and Appreciate. We use
some terms, concepts, and pedagogical ideas that we find extremely
useful that are not in most similar courses. Here are a few comments
about how they fit into the course.
Sections 1.1 and 1.2.
Section 1.1 Surveys the whole subject. So
that you can immediately talk real math to your students, it introduces
numerous useful terms, such as generalization and counterexample, even
though they will not be covered thoroughly until later. Then Section
1.2 on sets gives the students things to work with (concepts from set
theory)
that have terms that perfectly parallel the connectives they will study
in logic in the next sections.
Pronunciation and Grammar.
The
text
emphasizes
simple things that
other texts take for granted, such as correct pronunciation of symbolic
sentences. How do you pronounce “{x | x2 ≥ 4}”?
Students will not become comfortable with Mathematics if they cannot
even read the sentences! Many students hand in homework with symbols
combined in illegal ways (For example, “7 ⊂ {5, 6, 7}.”
We have many exercises that can be done quickly in class that help fix
these errors. To make the exercises easy for you to find during class,
they are marked with a smiley face, ☺.
☺ In the Homework Many
Simple Problems are Marked with this Symbol: ☺.
These problems are simple and very short. Nevertheless, they are
conceptually important. Going over problems aloud in class helps the
class learn these essential basics, and helps you, the instructor,
learn how uncomfortable or comfortable the students are. They can be
used anytime during class to make sure students "got it."
Many Activities for
Students. The book is designed so that there is a great
deal of illuminating work you can have the students do, even in class.
For example, in Chapter 3 on proofs there are many sample proofs for
students to learn from, but also many "conjectures" for students to
address. You might do a proof or two and set the students loose to work
on subsequence conjectures.
Conjectures.
One of our favorite teaching tools is the
conjecture. We put a mathematical sentence on the board explicitly
labeled “Conjecture” (e.g. "Conjecture: x < 5 =>
x2 <
25") and ask if it is “True or false?” (This one is false.
One counterexample is x = -10.) Usually the conjecture resembles
something true that they have seen. The text has many conjectures (both
in the exposition and in the homework), and many of them are false.
Conjectures help students learn to read with precision and to learn
that not everything is true. Conjectures are a great teaching tool to
teach critical thinking–to transfer responsibility for truth from
the authority (you) to the student.
Placeholders.
When letters are quantified in sentences they are
placeholders (also known as dummy variables) and the letters may be
switched (“2a + 3a = 5a, for all a” has the same meaning as
“2x + 3x = 5x, for all x”). When “S ⊂ T ”
is defined (Definition 1.2.12B), the definition tells us about subset,
but not about S or T. That definition also tells us when “P ⊂ Q” even though the
letters are different. Similarly,
the definition of “f is one-to-one” applies to “g°f
is one-to-one,” with the letters switched. It is critical that
students be able to switch letters when appropriate. We discuss
letter-switching early and often.
Naming Logical
Equivalences. In Sections 1.3-1.5 we give names
to
the most useful logical equivalences. For example, we call one of them
“Cases” because it refers to a common way to combine two
cases into one statement: (A or B) =>
C is logically equivalent to (A =>
C) and (B =>
C). Most proofs of a theorem in the first form
are
split into two cases–hence the name. You will find that being
able to look at a theorem and recognize its form (Oh! That’s
Cases!) is very helpful.
The only logical equivalences that we
consider are
those that are actually useful in higher mathematics. There are no
silly examples or artificially complex truth tables to create (The key
results are summarized on pages 86-88.)
Definitions in “Sentence
Form.” Proofs consist of sequences
of sentences. Logic applies to sentences, not to words. So, to do
proofs
we need to deal with sentences, not just words. We introduce the idea
of a defining terms “in sentence-form” (Definition 11, page
17, and again later). There is more than one way to define a term (for
example, we give three variants for some concepts such as set
intersection, page 16). Terms have a context, and putting them in a
sentence gives the context. In proofs involving new terms, we often
need to replace new terms with their definitions. They will need to be
in sentences, and we will show how sentences with new terms are
replaced with sentences with old terms. Students who learn
definitions “in sentence form” are more able to do proofs.
[Figure 4, Section 2.3, page 120.]
Concept Image and Concept
Definition. “Concept image”
and “concept definition” are two terms from mathematics
education that are relevant to instructing how to write proofs (Section
2.2, Definition 17, page 21, and Section 2.3, Definition 5, page 118).
Students often feel they know what something is without being able to
express themselves clearly when asked to define it. We all need and use
“concept images” which help us understand a term. But, when
it comes time to prove things, we need to know precisely what they are;
we need the “concept definitions.” By differentiating these
two levels of knowledge, we can describe to students the level they
need to attain to do proofs.
We Defer Proofs.
Proofs really begin in Chapter 3, although we do some
earlier. It takes quite a few preliminaries to understand proof.
Without them, proof is very hard to grasp, and experience shows that
most college sophomores do not have enough background in logic to be
ready for proof. (Most high schools no longer spend a year on
proof-based geometry; many students have very little exposure to
logical reasoning or proofs prior to this course.) With, and only with,
Chapters 1 and 2, the students are ready. The essential preliminaries
include: thinking about truth and falsehood, use of language and
especially placeholders, connectives and logic, common patterns of
reasoning, justification, and concept definitions. These are too often
neglected in other texts. All of these are discussed in Chapters 1 and
2 and are necessary prerequisites. You and your students will
appreciate all they learn in Chapters 1 and 2 before they are required
to do proofs.
Chapter
1
Section 1.1.
Preview of Proof
Why?
This section covers a great deal that will be covered
throughly in later sections [noted in brackets]. With these terms and
concepts you can discuss any part of logic whenever it comes up. The
students will at least have heard of it and you won't need
avoid math just because you have not lectured on the topic yet.
For example, truth-table logic usually
precedes the logic of quantifiers, but most mathematical conditionals
are generalizations (e.g. "If x > 2, then x2
> 4"). With the terms in Section 1.1, you can discuss
generalizations even before the sections devoted to them [Sections 2.1
and 2.2].
Mathematicians commonly think of
sentences with logical connectives in alternative forms (e.g. the
contrapositive is equivalent to the original conditional, Section 1.5).
Theorem 17 (A Hypothesis in the Conclusion) is an alternative form
which is very commonly used throughout mathematics (perhaps not with
this name). This theorem is followed by an example which shows how
theorems about "form" are used to reorganize theorems and their proofs.
[This continues in the Instructor's
Manual.]
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