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).
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
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into present or future lesson plans - some reading required.
Road
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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
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
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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
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areas [proportional amounts too] directly or by using maps and
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Mathematics
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Skills with take
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Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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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
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
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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