Proof: Introduction to Higher Mathematics
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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