Sets, Combinatorics and Probability Theory (136-148)
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Five Ways to Describe or Form Sets: The simplest way to
introduce sets is to give a finite list or roster of their
elements. Here is an example:
A = {2, 3, -5, 17, p}
In it, set elements are listed between left and right braces { and
}, a notational convention. With this list approach to sets, some
sets may be included in others - that leads to subset and superset
relations between sets. Operations of set complements, set
intersection, set unions and set products (the formation of ordered
pairs with first from one set and a second from another set, or the
same) may be introduced. The list approach to forming and defining
sets may appear early
If #A = the number of elements in a set A, then for two finite
sets A and B the law of inclusion-exclusion gives
That is easily explained and even used in junior high
school mathematics along with notation for set operations and
membership. The next simplest way to represent or define sets is
to draw Venn Diagrams, and in doing represent finite sets of points
and beyond that pictorially introduce sets of points in the plane.
That is easily explained and even used in level I mathematics along
with notation for set operations and membership.
Modern set theory (part of modern mathematics) assumes there are
three safe ways for describing and forming sets.
Set Applications:
Modern pure mathematics from counting to calculus may be formed and
expressed in terms of sets and set theory assumptions and
practices. The modern mathematics course designs of the 1950's
emphasized a introductory form of set theory and notation. Most
elements of mathematics were presented in terms of sets. Those
course designs lingers today in mathematics education as is or
diluted. The dilution was evident when I taught senior high school
courses whose content required set concepts to students whose
earlier education did not cover sets. That being said, set
concepts may be presence or useful in the description of
generalized grouping laws for calculating sums and products. Set
concepts (Venn Diagrams and subset builder notation) may be useful
in illustrating and developing logic mastery for the development of
mathematics and to aid precision in reading and writing outside of
mathematics. In counting for its own sake or the calculation of
probabilities, sets and functions may be employed to track and
count items, ways or possibilities. And probability itself may be
precisely expressed and calculated with the aid of set concepts.
Finally, the senior high school and calculus views of functions
and relations may employ a sets of ordered pairs to illustrate and
develop skills and concepts - whether that should be done in
general or case by case is open to discussion. In contrast to the
keen modern mathematics course designs of the 1950's, set skills
and concepts are most likely best introduced on a just in time
basis, just before or as they are needed and useful. The end of
senior high school preparation for calculus may for instance
include the modern mathematics high school level axioms, skills and
concepts in a formal manner, but reach that destination in manner
that makes the hard easier.
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Mathematical Induction and Recursive Definitions: Explain
factorial notation, summation notation and product notation. Prove
arithmetic and geometric summation formulas. Prove the Binomial
Theorem. Review the forward and backward use of geometric formulas in
the discussion of money matters.
Mathematical Induction, Inductive Definition, and
applications. Binomial Theorem. Geometric and Arithmetic Summation
formulas.
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Sets with Roster Notation (Grades 5 to 8). Use lists (roster)
notation to define sets and find their intersection, union and
complements with respect to each other.
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Sets and Plane Regions with Venn Diagrams (Grades 5 to
8): Illustrate set notation, unions, intersection and
complements with the aid of Venn Diagrams. Describe the intersections
verbally with the use of words AND, OR and NOT. Observe a set is given
by the complement of the complement of a set.
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Sets and Logic (Grades 9 or 10). Employ the Venn Diagram
viewpoint and roster or list viewpoint of sets to illustrate logical
operations and a meaning for the law of excluded middle statement (A
or NOT A) and the statement Not (A and Not A).
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Sets and Subset Builder Notation (Grades 8 or 9): Here what
has been called elsewhere set builder notation
B = { x in A | single or compound logical condition to be satisfied by
x }
will be introduced to build subsets B of a given set A. Mention in
passing the set of all subsets of a given set, and even a give a tree
diagram method for generating all subsets (proper and improper) of sets
with 2 to 5 elements.
Modern set theory (part of modern mathematics) assumes there are
three safe ways for describing and forming sets.
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Subset Builder Notation
Here one or more algebraic or logical conditions like f(x)
= c or alternatively, f(x) < c or f(x) < c
may be employed to form a subset of an existing set. Modern
set theory assumes the latter exist for any logical or
algebraic condition if A exists.
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Set Product Builder Notation:
The latter denotes the set of all order pairs (a, b) where
a comes from A and b comes from B. Modern set theory
assumes the latter exist if A and B exist.
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Power Set Builder Notation: - optional perhaps for high
school students:
The latter denotes the set P(A) of all proper and improper
subsets B of a given set A. Modern mathematics assumes P(A)
exists if A exists.
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Building Sets of Ordered Pairs (Grades 8 or 9): Introduce too
the algebraic assumption, that given two sets A and B (sets that may be
equal) then there is a set A × B of ordered based [a,b] where a belongs
to A and b belongs to B. The counting principle
#(A × B) = #(A) × #(B)
follows from the counting principles that #(A) rows of #(B) objects
gives #(A) × #( B) objects. The counting principle can be extended to
sets of ordered triplets and even quadruplets.
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Tree and Subtree Diagrams For Listing and Counting Possibilities
(Grades 8 or 9). Tree diagrams may be used to generate all
elements of a set products A × B, A × B × C, A × B × C × D and so on.
In the product of sets (options) case the product rules like #( A × B)
= #(A) × #( B), etc give the total number of possibilities listed and
generated. (Sub)Tree based counting can also be used for the recursive
identification of possibilities in set products A × B, A × B × C, A ×
B × C × D where for each element a of A, there are restrictions on
which elements b of B are permitted, where each admissible element
(a,b) of A × B, there are restrictions on the which elements c of C
are permitted. The former recursion may result in a subtree, one in
which the possibilities may be counted via addition without
multiplication shortcuts.
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Counting Elements in Disjoint Sets (Grades 8 or 9). If a set A
is the union of n-disjoint sets B j, then
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Law of Inclusion-Exclusion for Counting Elements in Overlapping
Sets (Grades 8 or 9). Use the law of
inclusion-exclusion
 in counting the number of elements in the union of
two sets A and B. Venn diagrams may be used to state and imply the law
of inclusion-exclusion for three overlapping sets.
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A Token or Ticket Method for Counting Elements in Overlapping Sets
(Grades 8 or 9). Imaging a group of people buy tickets. Let
Bj be the set of people that buy exactly j tickets. Then
the total number of people in the group is
In the union of n sets finite Aj ( 1 < j
< n), there the number of elements in the union is
if here Bj is the set of elements in the union that belong
to exactly j of the sets Aj. Imagine membership in each
Aj gives a ticket to each of its members. Note: the
token or ticket method here is an off-the-cuff observation (August 1,
2010). Most likely it re-invents or duplicates efforts elsewhere.
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Probability Theory (Grades 8 to 10). See and master the
expression and calculation of probability theory with the aid of sets.
Here function notation P(E) will be used to indicate the likelihood or
probability that an element of E will arise. There-in lies
opportunities to introduce combinatorial methods: employ factorial
function to count permutations, employ the binomial theorem and
coefficients to count combinations, and employ further set and
tree-based methods for listing and counting, that is enumerating
possible outcomes. There-in lies opportunities to introduce geometric
models for probability or hitting targets in the line, plane and
space. Conditional probability, tree diagrams for the latter, and
expected value of events or games may also be introduced. As usual all
rules and patterns may be used forwards and backwards.
Suggestion: Put the subset of probability with the most
accessible or greatest take home value first. Technical
Note: Combinatorics may employ set and functions and functions
of certain types as an aid to clarifying and counting
possibilities.
Remark: The division of probability theory over different years or
grade level is not specified here. Parts may come before and parts
after the finer discussion of functions and mathematical
induction.
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Sets of Numbers and Properties: Introduce notation for whole
and natural numbers, unsigned and signed rational numbers and real
numbers. The latter may be identified with coordinates along a
straight line and with finite and infinite decimal expansions, prefixed
by a sign. Discussion of terminating, periodic and infinite
non-periodic decimal expansions may suggest not all numbers are
rational. The number p and square roots of
primes may given without explanation why as the first examples.
Remark: The larger assumption that sums and products of signed
numbers may be calculated via addition of subtotals and multiplication
of subproducts includes and for most precollege students may
replace the need to describe commutative and associative properties
of addition and multiplication of unsigned and then signed real
numbers. With the product of signs being declared to be negative if an
odd number of factors are negative, and positive if an even number of
factors are positive, the property that products may be calculated
from subproducts follows the corresponding property for unsigned parts
and, a nuanced counting of factors with negative signs by subcounts.
Complex numbers through their introduction with rectangular and polar
coordinates for addition and multiplication inherit the assumed
subtotaling and subproducts properties from those for real numbers. By
mathematical induction, the distributive property of multiplication
over addition for complex numbers implies and justifies the use of
column methods for multiplication of two complex factors, one and then
both of which may be given by a sum. With mathematical induction, the
polar coordinate viewpoint of multiplication and the property that the
product of two unsigned numbers are nonzero if the factors are nonzero,
the product of complex number factors -two and more - must be nonzero
if all the factors are nonzero. It easily shown that complex numbers
have additive inverses and multiplicative inverses. So subtraction and
division are possible and may be described via the addition and
multiplication of those inverses.
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Function or calculation rule viewpoint Arithmetic Properties of Real
and Complex Numbers and arithmetic identities. Algebraic
expressions for associative properties of addition and multiplication,
the distributive law a(b+c) = ab+ac imply different calculation rules
f and g say give the same value for the same set of arguments. The
assumption that two different methods f(z1, z2,
z3) and g(z1, z2, z3) give
the same value for a sum or product for all triplets of numbers
z1, z2, z3 in a set S may phrased in
terms of equality of functions or calculation rules. So we may write
f(z1, z2, z3) = g(z1,
z2, z3) for all numbers z1,
z2, z3 in that set S. Thus two calculation
rules are equivalent - they give the same function. (Pure
mathematics may have to identify functions with an equivalence class or
in the case of piecewise defined functions, a manifold collection of
subfunctions or calculation rules where the latter agree on their
common domain) Similar remarks follow for algebraic expressions in
two variables - those say that arise from the commutative properties of
addition and multiplication.
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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
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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
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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
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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Mathematics
Skills For Ages 3 to 14 - technical!
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
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Arithmetic
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Algebra
Starter Lessons
Geometry
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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