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Home < Volume 2 Three Skills For Algebra << Chapter 30 Truth Tables

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Chapter 23, Truth Tables

Introduction

As a student I was never satisfied with the explanation or justification of entries in truth tables for material implication given in my courses. Sp here is an alternative. The occurrence table in chapter 21 for B IF A will be used to explain or provide a justification for truth tables for material implications B IF A (or equivalently, IF A THEN B). Truth Tables appear in upper high school and in college mathematics as an echo of modern notation the algebraic (or symbolic) study and codification of logic. Truth tables may also appear in the discussion of logic tables for electronic circuits: AND, OR, NOR and NAND. Hence, truth tables appear in school mathematics and electricity related courses. Truth tables are useful for showing the equivalence of an implication with its contrapositive form. Truth Tables stem from the work of the philosopher Ludwig Wittgenstein.

Instead of talking about rules and situations (or events) we will talk in this section about statements and assertions. Suppose A and B are shorthand symbols for statements (events, situations etc.) which can be true or false but not both simultaneously in a given situation. Given two such statements A and B, we can define the new statements A or B, A and B, if A then B, NOT A and A iff B. Our goal in this chapter is to say when these new statements are true and when they are false.

The foregoing phrases in terms of situations and rules can be expressed as follows:

  • a statement of the form A or B is true when at least one of the statements A and B is true. Otherwise it is false.
  • a statement of the form A and B is true when both of the statements A and B are true. Otherwise it is false.
  • a statement if A then B is declared to be true if (i) statement B is true whenever statement A is true and (ii) whenever statement B is false, so is statement A.
  • a statement NOT A is declared to be true when A is false, and this statement NOT A is declared to be false when A is true.
  • when at least one of the statements A and B is true, so is the other, and provided (ii) that when at least one of the statements A and B is false, so is the other. (All this is a bit of a tongue twister.)

NOT Revisited

The following truth table shows the relationship between the truth (T) and falseness (F) of A and NOT (A).

row A not(A)
1 T F
2 F T

The statement A is always true when statement NOT A is never true.

The statement NOT A is always true when statement A is never true. Here instead of saying never true, we may say always false.

AND Revisited

The truth (T) or falseness (F) of the statement A and B depends on the respective truth or falseness of the statements A and B. This situation is summarized in the following table.

row statement A statement B A and B
1 T T T
2 T F F
3 F T F
4 F F F

The statement A and B is said to be always true (to always hold) if the situations in rows 2, 3 and 4 of the above table never occur.

OR Revisited

The statement A or B is said to be (mathematical usage) when and only when at least one of the statements A and B is true. The following table summarizes this situation. It shows when the statement A or B is true and when it is false.

row statement A statement B A or B
1 T T T
2 T F T
3 F T T
4 F F F

With this usage, the statement A or B is guaranteed to be true provided the situation in row 4 of the above table never occurs.

If-Then Revisited

We consider the implication if A then B. The following table signals when this implication rule is false and when it is true. Here false signals the rule implication is disobeyed, while true signals not disobeyed. We declare that an implication rule if A then B is always true provided the situation in row 2 never occurs.

row statement A statement B if A then B
1 T T T
2 T F F
3 F T T
4 F F T

The implication if A then B is said to be vacuously true when statement A is always false.

If-and-Only-If Revisited

The following truth table if for the two-way implication A if and only if B. We observe the two-way implication is always true if the situations in rows 2 and 3 never occur.

row statement A statement B A if and only if B
1 T T T
2 T F F
3 F T F
4 F F T

Remember the letter F signals false, and corresponds to the idea of rule being disobeyed. Also remember that the letter T signals true and corresponds to the ideas of a rule being obeyed, or not disobeyed.

Teachers & Tutors: Site pages offer better or best practices for providing skills - simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do. Others are welcome to refine or exceed it. Please do.

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

The Logic of Injustice: How Texas sent an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate is not compensation for years or decade of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
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5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.


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Home < Volume 2 Three Skills For Algebra << Chapter 30 Truth Tables

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34][35] [36] [37] [38] [39] [40] [41] [42]


Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
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2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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