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Thu Oct 11 14:06:13 PDT 2007

## Why don't you have the answer to my question

I'm sorry that I have failed you.

This site started out as a collection of fundamental mathematical and logical systems that would help people develop software. I didn't put in many things because I didn't think they would be needed.

Later I added some answers to question and pieces of logical mathematics that people asked me. I've never seen your question before or decided it was outside the scope of this site. Your answer may very well be on one of the sites below.

## Where can I find simple mathematics on the web

Try this site [ http://math.com ] (with commercials) or [ math-faq.html ] for a start.

The Wikipedia [ Main_Page ] is a pretty reliable source of mathematics.

## What are the 3 operators of Set Theory

The three set operations are (normally)
1. Union
2. Intersection
3. Complement.

If you have a set of red things and a set of rectangles than their intersection is the set of red rectangles. Their union is the set of red things plus the set of rectangles. The complement of the red things with the rectangles are all the non-red rectangles.

Symbolism:

• Union -- a cup shaped wingding
• Intersection -- a cap shaped wingding
• Complement --- various including A\B.

Formal

(intersction): A ∪ B = { x. x in A or x in B }.
(union): A ∩ B = { x. x in A and x in B }.

For more see [ intro_sets.html ] , [ logic_30_Sets.html ] and [ logic_32_Set_Theory.html ]

## What do the following terms mean?

1. congruence::geometry="Two figures are said to be congruent if they can match perfectly by sliding one, rigidly, on top of the other". Congruence is geometrical equality. Two triangles are congruent if they have the same angles and sides. Indeed if they have the same sides (same lengths) they are congruent. If the have two angles and one side equal then they must be congruent. Finally, if they have two sides and the angle between the sides is equal then they are congruent. Notice that if their are two equal sides and an equal angle that is not between them then they may be two different triangles.
2. substr::=Substring operator in BASIC, awk, C, C++, ... A string s is a substring of string t if and only if there are two strings x and y (which can be empty) and x s y = t.

3. congruence::modern_mathematics=Any equivalence relation that also respects a set of operations and/or functions.

4. sinx::=Casual mispelling of sin(x), the sine function,...
5. sin(x)::=a function created to solve problems in measuring triangles that turned out to be very useful in many branches of mathematics. If you leaned a ladder against a vertical wall with the base of the ladder on the horizontal ground and the ladder is l feet long and the angle at the ground is x then the ladder reaches l*sin(x) up the wall. It is also l*cos(x) from the wall. Many waves are said to be sine waves because their height varies like sin(x).

6. relational algebra::=Any algebra that has the same laws as Binary Relations. [ intro_relation.html ] [ logic_40_Relations.html ] [ logic_41_HomogenRelations.html ] [ logic_42_Properties_of_Relation.html ]

7. regular_expressions::=Expresion made of sequence, selection and iteration. [ logic_41_HomogenRelations.html ] [ math_61_String_Theories.html ] [ math_62_Strings.html ] [ math_63_Languages.html ] [ math_64_Meta_Macros.html ] [ math_71_Auto...Systems.html ]

8. reflexivity::=Property of relationships. If R is reflexive then x R X is true for all x. (=) ==> R. [ intro_relation.html ] [ logic_40_Relations.html ] [ logic_41_HomogenRelations.html ] [ logic_42_Properties_of_Relation.html ]

9. set_theory::=The mathematics of collections of objects, For more see [ intro_sets.html ] , [ logic_30_Sets.html ] and [ logic_32_Set_Theory.html ]

10. thearom::misspelled=theorem
11. Pythagorean_Theorem::=The sum of the squares on the two sides of a right triangle equals the square on the hypotenuse.

12. pi::=A greek letter, normally means the ratio between diameter and circumference of a circle, turns up all over the place in math. It is approximately equal to 3.141592653589793... . It is usually written π. The Wikipedia [ Pi ] article is worth visitting.

13. phi::=A greek letter, normally means a constant, the ratio that turns up all over the place in recreational math. Related to the Golden Rectangle. In Greek the symbol is φ.

14. one_one_and_onto::=A kind of map/morphism/relationship. f is X---Y if every x:X maps into precisely one y and every y has precisely one x. [ intro_function.html ] [ logic_5_Maps.html ]

15. linear_programming::=Linear_Programming.
16. Linear_Programming::=A technique for finding a set of values for variable that optimize a linear function subject to linear constraints. [ partVIII.htm ] [ tutorials.html ] (With thanks to Dr. Keith Schubert of CSUSB).

17. linear_equations::=A set of equations where the unknowns are multiplied by given constants and added up.

18. distribution::=In statistics, the relative chances/frequencies of different events occurring.

19. C::=A constant, or a programming language.

20. box_and_whisper::=Mispelling of
21. Box_and_whisker_diagram::=A Statistical diagram that has a box for where most of the data is and whiskers showing the tail. Tukey

22. lattice::algebra=an algebra with two operators that generalizes boolean algbras by having fewer assumptions, [ Lattice in math_41_Two_Operators ] for the formal definition. Lattices are rather elegant algebraic structures, and they were used to develop denotational semantics of programming langugaes in the middle 1970's.

23. ontology::=A description of how ideas in a domain fit together in a logical structure. Typically an ontology is a list of words that refer to concepts or sets of objects. Add to this relations between them such as: "A is a special kind of B", "A is a part of B", "A is not a B", etc. Here are some samples [ logic_8_Natural_Language.html ] of some ontologies for English taken from several sources and expressed in my MATHS notation. Ontologies are an active research area since one part of an object-oriented system should be a running model of some real objects... and the structure follows the ontology of reality.

(Mathematical FAQ): [ math-faq.html ]
(Mathematical logic on the web FAQ): [ world.html ]

. . . . . . . . . ( end of section FAQ Frequently Asked Questions) <<Contents | End>>

# Notes on MATHS Notation

Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

# Notes on the Underlying Logic of MATHS

The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.

For a more rigorous description of the standard notations see

1. STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html

# Glossary

2. above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
3. given::reason="I've been told that...", used to describe a problem.
4. given::variable="I'll be given a value or object like this...", used to describe a problem.
5. goal::theorem="The result I'm trying to prove right now".
6. goal::variable="The value or object I'm trying to find or construct".
7. let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
8. hyp::reason="I assumed this in my last Let/Case/Po/...".
9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.