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YOU are better than YOU think. Show yourself how: |
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Logic
Mastery Do not leave here without it - 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 linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; 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. |
Functions and Sets
1 Why Set Theory
Set membership, union, intersection and complements form a language for a precise description of mathematical ideas.1 Set theory is emphasized in mathematics since the axiomatic method of analysis, more precisely the arithmetic properties of integers, real numbers and functions can be logically codified and described in it. Outside calculus, you may see the role of set concepts in some presentations of analytic geometry. Lines, planes, surfaces and sometimes solid objects are regarded as set of points. Sets and set membership also have a role in combinatorics and counting, and in the combinatorial based parts of probability. Combinatorics counts or is concerned with the number of ways objects in sets can be grouped or placed together.
2 Set and Rule Viewpoints Reconciled
When two different views of an idea are presented, they need to be reconciled. In the first instance, we could regard a function as a well-defined computational rule on a domain. The rule could be given by a formula, and the domain might be defined to the maximal set of real (or complex) numbers for which division by zero and other undefined operations are avoided. A set of ordered pairs which satisfies the vertical line property could be viewed as just another (and then a more general way) of representing or defining a function. Here
- a formula y = f(x) (or rule for computing y given some x in a domain) can be used to define a set of order pairs with the vertical line property, and
- a set with the vertical line property can be used to define a rule for computing y when x lies in the domain of the set.
The foregoing links together and treats equally the set-theoretic definition of functions and the rule-based definition. The issue of which rule is better is not important except in the set theoretic foundations of mathematics. Both definitions should be viewed as interchangeable.
3 Relations
The use of the word relation in set theory requires some explanation. An explanation is given here since, as suggested earlier, definitions and theories without examples to illustrate them hold vacuously. Now requiring that an equation be satisfied by (x,y) when giving the value of x restricts the value of y, and vice-versa. The satisfaction of an equation by two quantities (x,y) defines what is called a relationship between them.From equations to sets. For instance observe graphing and sketching the solutions of an equation, for example y2+2x2 = 1, defines sets of ordered pairs (a solution set) in the plane. Requiring that an equation be satisfied by the ordered pair (x,y) is equivalent to requiring that this ordered pair belongs to the solution set of the equation.
In set theory, a set S of ordered pairs is said to be a relation: The knowledge that a point (x,y) belongs to a set S links (relates) the values of x and y. In particular, for a set S of points in the plane, giving the value of x restricts the value of y to the intersection of the set S with a vertical line through the point (x,0) while specifying the value of y restricts the value of x to the horizontal line through the point (0,y). This is exactly what happens when S is a solution set for an equation.
Back to an Equation. Note given a set S of points, we define the characteristic function XS of the set by putting XS(x,y) = 1 whenever (x,y) belong to the set S, and putting XS(x,y) = 0 otherwise. Now the set S is the solution set of the artificial equation XS(x,y) = 1. So we can go back to the equation viewpoint of relations.
Footnotes:
1But it is not
always employed in mathematics instruction. In college-level calculus texts, the
set theoretic representation of functions may be mentioned or emphasized for
function of single variable, but it is frequently omitted in the description of
functions of several variables to science and engineering student for whom
specialization in math is no longer of interest.
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Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.