YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
Definition via infinite sets of ordered pairs or curves in the plane
Vertical Line Rule and Method
![[sketch of a graph and illustration of vertical line rule]](images/exampl2.gif)
[A set of points]
In the above diagram, we see each vertical line touches the drawn set or curve at most once. This gives us the computation rule: for each real number x for which the vertical line through x has a unique intersection (x,y), we put f(x) = y and then write y = f(x).
Failure of the Vertical Line Rule and Method
For the above diagram, for some values of x, the vertical line through x meets the set S of points at more than one point (x,y). That is there are several values of y and not a unique value for which (x,y) belongs to the set. So for this set S, we can not use a vertical line to find a unique y given x. Thus no computation rule for y given x can be defined. But if we know (x,y) belongs to S, we can say that y belongs to the vertical line with horizontal coordinate x. That relates or links the value of y to the value of x.
The Set Viewpoint
Skip on first reading. In later sections, we will meet the following definitions and context for them.
Relation: Suppose A and B are sets of objects (numbers, points in the plane, etc).. Suppose F is a subset of the product set
A x B = {(a,b) | a belongs to A and b belongs to B}
Then F is called a relation. In other words, any set of order pairs is called a relation.
Now suppose F is a relation. If we say (x,y) belong to S then y must belong to the set
F(x) = { (x,b) in A x B | (x,b) belongs to S}
If the latter set has only one value (or none) for all x in A, we say y is a function y =f (x) of x and the domain of this function is the set of points x for which F(x) consists of only a single point. When F(x) has only a single element, that element is the value of f(x).
Remark 1: The equation x = y2 says how to compute x from y. So x = g(y) is a function of y. However, if we graph the set of points x = g(y) = y in the coordinate plane we get a parabola with the x-axis as its axis of symmetry.
When a > 0, the vertical line x = a intersect the curve at two points, namely (a, sqrt(a)) and (a, -sqrt(a)). So the equation x = y2 or a = y2 has two solutions y = sqrt(a) and y = -sqrt(a). Here the equation limits the values of y when x = a, but does not determine the value y. So y is not a function of x. However y is related to x =a (do not think of sets when this word related is used) by the requirement that y = sqrt(x) or y = -sqrt(x). Before set theory, two real variables x and y were related if they were required to satisfy an equation. With the advent of set theory, the solution set of an equation gives a set of ordered pairs in the plane. The set theory codification of what is a relation keeps the set of ordered pairs in the plane but does not require that set be the solution set of an equation.
Remark 2. The equation y = x2 says how to compute y from x. So x = g(y) is a function of y.
For all real numbers a, the vertical line x = a has only when intersection point (x,y) = (a, a2) with the set of order pairs in the graph of . So that set of ordered pairs is not only a relation in the set-theoretic sense between x and y, it is also a relation F which satisfies the vertical line property. So if the equation y = x2 was forgotten or not available for computation, we could still in principle use the above graph and the vertical line rule (method?) to find y from x.
ADVERSTISEMENT
www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelFNs & Dependency FN With Finite Sets FN Vertical Line Rule FN Infinite Domains FN Sets-Theory FN Interval Notation FN: Sets - Continued (FN) Sets & Relations I (FN) Relations & Sets FN Source Target Domain Range (FN) Injective or Not (FN) Sign & Zero Analysis (FN) Increasing/Decreasing (FN) Extrema FN Numerical Exercises FN Step Sawtooth Abs.Value FN Horizontal Line Rule FN Inverse Functions FN Many Ways to Define
Area Entrance (C) Complex Numbers (FN) What are Functions? (FN) Functions - More SZM: Sign Analysis (L) Lines Summary (P) Polynomials (*,+,-) (Q) Quadratics (D) Simplify Square Roots (T) Unit Circle Trig Conic Sections Links More Links
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse Real Numbers Say More Positive Linear Inequalities Triangle Inequality Absolute Value |x| |x| Eq'ns & Inequalities Rectangular Coords Shortest Path Distance Formulas Add & Multiply Points Polar Coordinates Radians (A) Vectors (A) Coordinate Arithmetic (A) Navigation on Maps (A) Addition Geometrically (A) Rotation (PT) Translations (PT) Dilatations PT: Rotations
Links to Site Pages outside this site area follow - co- and pre- requisites.
Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!