19. Symbols & Logic

Chapter 19
Elements of Logic

Logos

thought.

Yet in mathematics, direct and indirect chains of reasoning appear. The study of logic, that is, methods or laws for rule- and pattern-based thought, has been motivated by the need in mathematics to reach conclusions. In particular, proofs based on (1) mathematical induction, (2) the contrapositive, and (3) proof by contradiction all stem or originate from the conclusion-reaching needs of mathematics.

The chapter Direct and Indirect Reason below, will describe methods (2) and (3). Suggestion: try to read this last chapter to see how much can be immediately understood.

The subject of logic as it is studied within mathematics courses, is often presented as an algebraic (or symbolic) perspective of the methods of reason. The next chapters present the algebraic perspective. They with the earlier algebra-free discussion of implication rules and chains of reason give some preparation for the description of the indirect methods (2) and (3) for in the last chapter Direct and Indirect Reason

The algebraic description of logic also has a role in the design and simplification of electrical controls and computing circuits.

The algebraic description of logic further allows algebraic methods for arriving at conclusions, in particular mathematical induction, to be applied to the drawing conclusions about rule-based reason and logic. The algebraic description of logic provides models of mathematical logic. Conclusions drawn about the models then reflect on the limitations and reach of logical or rule-based thought in mathematics.

1 About the Next Chapters

The next five chapters

continue the description of logic.

The occurrence (or obedience) tables invented and introduced below identify those situations in which implication rules are obeyed, disobeyed or not disobeyed. The latter notions are intended to simplify the explanation of truth tables. An implication rule is said to be true in the case when it is obeyed or it is at least not disobeyed. An implication rule is said to be false or not true when it is disobeyed.¹

The chapter The Contrapositive shows the equivalence of an implication rule with its contrapositive formulation. The analysis is based on the three notions of a rule being obeyed, disobeyed or not disobeyed.

The chapter Direct and Indirect Reason describes and explains direct and indirect methods for reaching or proving conclusions. Among the indirect methods, this chapter describes in particular, how an implication rule can be shown to always hold by (a) showing its contrapositive form always hold, or by (b) looking for absurdities that would occur if the implication rule did not hold. The second method (b) is more indirect than the first method (a).

¹ The language previously used to explain and justify the entries of truth tables overuses the word true. The introduction of the three notions of an implication rule if A then B being obeyed, disobeyed or not disobeyed aims to avoid this situation. Such implication rule is said to be false in situations where it is disobeyed, and it is said to hold (or be true) in those situations where it is obeyed or at least not disobeyed. Finally, the implication rule is said to be always true in the circumstances of interest provided it is never disobeyed in those circumstance. See the text for further explanation.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Chapter 19 Elements of Logic

1 About the Next Chapters

Chapter 19
Elements of Logic