Pigeon Hole Principles
Appendices, Volume 3, Why Slopes and More Math.
The Finite Case
The finite pigeon hole principle is given by the contrapositive of the
following implication: If there is no more than one pigeon in each of
$n$ holes, then the $n$ holes contain at most $n$ pigeons. The
equivalent contrapositive of this implication follows. If $n$ holes
contain more than $n$ pigeons, then at least one hole contains more than
one pigeon. Here it is possible to examine each hole or location, one
at a time. After at most $n$ inspections, a first hole with more than one
pigeon will be found. Instead of pigeons and holes, we could speak of
elements and sets in accordance with the set-theoretic formulation
favored in the axiomatic formulation and codification of modern
mathematics.
The Infinite Case
More generally, the infinite pigeon hole principle is as follows. If
$n$ holes contains an infinite number of points (small pigeons) then at
least one of the holes contain an infinite number. In particular, if
the holes are labeled or ordered from 1 to n, then there must be a first
hole with infinitely many points (small pigeons) in it. But a finite
number of inspections need not say which one. The infinite pigeon hole
principle is the contrapositive of the following implication rule: If
each of $n$ holes contains finitely many pigeons then all $n$ holes
together contain finitely many pigeons.
Imagine for instance that we try to inspect the holes one at a time in
sequence in the hope of identifying the first hole with infinitely many
points. Further suppose that the points in a hole can be counted one per
second. Now if a hole has finitely many points, we can count them all in
a finite time. But if a hole has infinitely many points, counting them
one per second will begin but never end. Even after counting a large
number of points in the hole, there still may be a small, large or
infinite number of points in the hole. So even if a count does not appear
to be ending, we cannot say that it does.
Food for thought:
In principle perhaps, time and temporal notions have no place in the
comprehension or visualization of mathematics: we can see or
understand everything at once. But logical and numerical reasoning
or computations as they are followed or done depend on a sequence of
operations or counts in time. In particular, mathematical reasoning takes
time to be done or followed. Yet with hindsight, that reasoning and the
results obtained can be reviewed and understood with a fleeting thought ---
divorced from the passage of time. The reasoning and the results appear to
have an objective (Platonic) existence apart from the passage of time. To
say more would delve into several philosophies of mathematics and logic.
|
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
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
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
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.
|
|