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Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices. Site's Best LessonsFor Logic
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. For Arithmetic
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style. For Algebra
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct? |
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 30 Truth Tables Next: [Chapter 31 Direct-and-Indirect-Reason.] Previous: [Chapter 29 Contrapositive and Vacuously True Implications.] [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] Chapter 23, Truth TablesIntroduction
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:
NOT RevisitedThe following truth table shows the relationship between the truth (T) and falseness (F) of A and NOT (A).
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 RevisitedThe 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.
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 RevisitedThe 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.
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 RevisitedWe 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.
The implication if A then B is said to be vacuously true when statement A is always false. If-and-Only-If RevisitedThe 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.
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. Postscript: Above the statement if B then A has the same meaning as the statement A if B. When the latter statement is true, we know that when B is true, then A has to be true as well. But when A is true, we do not know that A has be true. However, if we are given the statement A if and only if B, the latter again requires that A be true when B is, but also it goes further say that A can be true only when B is. Selby A, Volume 2, Three Skills for Algebra, 1996. www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 30 Truth Tables Next: [Chapter 31 Direct-and-Indirect-Reason.] Previous: [Chapter 29 Contrapositive and Vacuously True Implications.] [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] |
Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic? Site Reviews1996 - Magellan, the McKinley Internet Directory: Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it. 2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
For Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale. For Calculus
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. |