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1. Probability, Random Variables and Stochastic Processes with Errata Sheet by Athanasios Papoulis, S. Unnikrishna Pillai, 2001-12-14

1. Saul Landau: Lying And Cheating, Bush's New Political Math
America's Best Political Newsletter. Does your "new political math" rest on different axioms than the old political math, the teenager inquired? Not too different, I said. math axioms apply universally. But this axiom
http://www.counterpunch.org/landau11292003.html
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2. Canadian Dimension: LYING AND CHEATING: BUSH'S NEW POLITICAL MATH
Does your new political math rest on different axioms than the old political math, the teenager inquired? math axioms apply universally.
##### LYING AND CHEATING: BUSH'S NEW POLITICAL MATH
By Saul Landau
November 2003 As a parent, I faced the challenge of helping my kids learn the new math. But how do you help kids grasp contemporary reality when the equivalent of the cognitive shift in numbers hits politics? She asked for help in her Poli Sci course.
##### BUSH'S NEW POLITICAL MATH
"The de facto role of the United States Armed Forces will be to keep the world safe for our economy and open to our cultural assault. To those ends, we will do a fair amount of killing." Does your "new political math" rest on different axioms than the old political math, the teenager inquired? Not too different, I said. Math axioms apply universally. But this axiom supposes that the United States bombing raids, invasions and occupations of foreign countries derives from God-driven motives. Oh, so when Saddam and other evil men kill, it's not for God, but to satisfy their personal power lusts? Yes, I said, you learn fast. And post war contracts of billions of dollars to Halliburton, VP Dick Cheney's old company, do not appear in this new axiom; nor does the fact that Iraq grows oil, not apples, she added. Right. Now, go to Sicily for the next piece of the new math. The "You can say anything lemma," derives from an aged Mafia Don complaining to his doctor about his 95 year old cousin's boast of having frequent intimate relations with a young woman.

3. Origami & Math
Wolfram Research has a page about origami and math which has pictures, lists Huzita s axioms, and provides many further references.
http://www.paperfolding.com/math/
 So, you're interested in origami and mathematics...perhaps you are a high school or K-8 math teacher, or a math student doing a report on the subject, or maybe you've always been interested in both and never made the connection, or maybe you're just curious. Origami really does have many educational benefits . Whether you are a student, a teacher, or just a casual surfer, I have tried my best to answer your questions, so please read on. So exactly how do origami and math relate to each other? The connection with geometry is clear and yet multifaceted; a folded model is both a piece of art and a geometric figure. Just unfold it and take a look! You will see a complex geometric pattern, even if the model you folded was a simple one. A beginning geometry student might want to figure out the types of triangles on the paper. What angles can be seen? What shapes? How did those angles and shapes get there? Did you know that you were folding those angles or shapes during the folding itself? For instance, when you fold the traditional waterbomb base, you have created a crease pattern with eight congruent right triangles. The traditional bird base produces a crease pattern with many more triangles, and every reverse fold (such as the one to create the bird's neck or tail) creates four more! Any basic fold has an associated geometric pattern. Take a squash fold - when you do this fold and look at the crease pattern, you will see that you have bisected an angle, twice! Can you come up with similar relationships between a fold and something you know in geometry? You can get even more ideas from this presentation on

4. Math Forum: Discussion Search Results
1 match) Creative math.;w/axioms of Time and Whole Values Re Modern math. axioms based on Sheep counting axioms ( 2 matches)
http://mathforum.com/~richard/zim.olson.html
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5. Geometry: Axioms And Postulates
1.3 The Substitution Axiom. The third major axiom is the substitution axiom. It seems natural enough, but is necessary to form the foundation of higher math.
http://www.sparknotes.com/math/geometry3/axiomsandpostulates/section1.html
 Advanced Search FAQ document.write("My Preferences"); My Preferences Related Message Boards Intriguing Problems Trigonometry Algebra Geometry We want your feedback! Please let us know if you have any document.write("comments, "); document.write("requests, "); document.write("or if you think you've found an "); document.write("error."); comments requests , or if you think you've found an error Watch us work! Home Free Study Aids ... Geometry: Axioms and Postulates Axioms of Equality - Navigate Here - Summary Terms Axioms of Equality Axioms of Inequality Postulates Axioms of Equality In this section, we will outline eight of the most basic axioms of equality. The Reflexive Axiom The first axiom is called the reflexive axiom or the reflexive property. It states that any quantity is equal to itself. This axiom governs real numbers, but can be interpreted for geometry. Any figure with a measure of some sort is also equal to itself. In other words, segments, angles, and polygons are always equal to themselves. You might think, what else would a figure be equal to if not itself? This is definitely one of the most obvious axioms there is, but it's important nonetheless. Geometric proofs , as well as proofs of all kinds, are so formal that no step goes unwritten. Thus, if perhaps two triangles share a side and you wish to prove those two triangles congruent using the SSS method, it is necessary to cite the reflexive property of segments to conclude that the shared side is equal in both triangles.

6. Geometry: Axioms And Postulates
axioms and Postulates Two of the most important building blocks of geometric proofs are axioms and postulates. In the following
http://www.sparknotes.com/math/geometry3/axiomsandpostulates/summary.html
 Advanced Search FAQ document.write("My Preferences"); My Preferences Related Message Boards Intriguing Problems Trigonometry Algebra Geometry We want your feedback! Please let us know if you have any document.write("comments, "); document.write("requests, "); document.write("or if you think you've found an "); document.write("error."); comments requests , or if you think you've found an error Watch us work! Home Free Study Aids ... Geometry: Axioms and Postulates Axioms and Postulates - Navigate Here - Summary Terms Axioms of Equality Axioms of Inequality Postulates Axioms and Postulates Two of the most important building blocks of geometric proofs are axioms and postulates . In the following lessons, we'll study some of the most basic ones so that they will be available to you as you attempt geometric proofs. Axioms and postulates are essentially the same thing: mathematical truths that are accepted without proof. Their role is very similar to that of undefined terms : they lay a foundation for the study of more complicated geometry. Axioms are generally statements made about real numbers. Sometimes they are called algebraic postulates. Often what they say about real numbers holds true for geometric figures, and since real numbers are an important part of geometry when it comes to measuring figures, axioms are very useful. Postulates are generally more geometry-oriented. They are statements about geometric figures and relationships between different geometric figures. We've already studied some, such as the

7. Zermelo-Fraenkel Axioms -- From MathWorld
search. Abian, A. On the Independence of Set Theoretical axioms. Amer. math. Monthly 76, 787790, 1969. Devlin, K. The Joy of Sets
http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Axioms Foundations of Mathematics ... Szudzik Zermelo-Fraenkel Axioms The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory stands for exists for does not exist, for "is an element of," for the empty set for for all for implies for NOT negation for AND for OR for "is equivalent to," and A y ) denotes a formula of a set x consisting of all elements of a satisfying A y Axiom of extensionality Note that some texts, such as Devlin (1993), use a bidirectional equivalent while others, such as Enderton (1977), use the one-way implies One-way implication suffices. Axiom of the unordered pair Axiom of the sum set Axiom of the power set is confusing, and possibly incorrect. Axiom of the empty set Axiom of infinity (Enderton 1977).

8. Hilbert's Problems -- From MathWorld
geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and math.
http://mathworld.wolfram.com/HilbertsProblems.html

9. Sci.math Topic
All Discussions sci.math Topic. Date. Topic. Author. 07 Dec 03. axioms of set theory (Comprehension vs Pairing) Imam Tashdid ul Alam. 07 Dec 03. Re axioms of set theory (Comprehension vs Pairing) William Elliot
http://mathforum.com/discuss/sci.math/t/560781
 The Math Forum sci.math Search All Discussions sci.math Topic Date Topic Author 07 Dec 03 Axioms of set theory (Comprehension vs Pairing) Imam Tashdid ul Alam 07 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) William Elliot 07 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Marco Lange 07 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Arturo Magidin 08 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Eli 08 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Torkel Franzen 08 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Thomas Bushnell, BSG 09 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) William Elliot 09 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Imam Tashdid ul Alam 09 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Thomas Bushnell, BSG 11 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Imam Tashdid ul Alam 11 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Thomas Bushnell, BSG 13 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Herman Rubin 14 Dec 03 Re: Axioms of set theory (Comprehension vs Pairing) Thomas Bushnell, BSG

10. Peano's Axioms.
Next Natural numbers. Peano s axioms. When Peano created his axioms he wanted to catch the spirit of the natural numbers is a small set of rules.
http://hemsidor.torget.se/users/m/mauritz/math/num/peano.htm
 Created 980625. Last change 980728. Previous : Numbers . Up : Contents . Next : Natural numbers Peano's Axioms. The first part of these pages about numbers will use the Peano's axiom system as a foundation. The arithmetic created using this is called Peano Arithmetic, PA . Further down will you find a look at the numbers defined using the The Zermelo-Fraenkel's system, ZF When Peano created his axioms he wanted to catch the spirit of the natural numbers is a small set of rules. He created a starting point by axiom number 1, a way to get more naturals by axiom 2, a way to ensure that really is the starting point by axiom 5, and so on. A informal way to write these axioms could be : 1 : is a natural number. 2: If a is a natural number then so is a+1. 3: If you can prove something about a and that implies that you can prove it for a+1, and if you can prove the very same thing for , then will this hold for all natural numbers. 4: If a+1=b+1 then a=b.

 11. Math 4606, Summer 2002 Inductive Sets, N, The Peano Axioms math 4606, Summer 2002 Inductive sets N, the Peano axioms, Recursive Sequences ver The equation only uses set axioms and the axioms for R.Since every inductive set musthttp://www.math.umn.edu/~jodeit/course/InductiveSets.pdf

12. In A Few Words...
author, Abu Ja far Mohammed ibn al Khowarizmi (circa 825) who wrote a math textbook To define it rigorously we may need a set of axioms, like those proposed by G
http://www.cut-the-knot.org/do_you_know/few_words.shtml
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##### In a few words...
While putting together these pages I sometimes feel a need to refer to a term without straying from the topic at hand. Oftentimes it's possible to locate a resource on Internet with a necessary definition but cumbersome to specify the reference. In short, I decided to maintain a page of very short topical descriptions which, if and when a need or inspiration induce me to, I'd be able to expand upon. Many of these have been mentioned on the Did you know... page where, as a group, they, I hope, provided some entertainment. As I had neither immediate need for nor intention to describe them, some terms from that page have been left dangling without any reference or definition. Hence the current page.
• Absolute value
The absolute value is defined for real
• Algorithm
The word algorithm comes from the name of a Persian author, Abu Ja'far Mohammed ibn al Khowarizmi (circa 825) who wrote a math textbook. The word refers to a precise prescription (given by a step-by-step description) of a solution to a problem.
• Braids Theory
Braids Theory was invented by Emil Artin and is a part of the Knot Theory.

 13. Math 4606, Summer 2002 Inductive Sets, N, The Peano Axioms math 4606, Summer 2002 Inductive sets N, the Peano axioms, Recursive Sequences v. 1.1 Page 1 of 10Inductive sets (used to dene the natural numbers as a subset ofR)( 1) DenitionA set S The equation only uses set axioms and the axioms for R.Since every inductive set must containhttp://www.math.umn.edu/~jodeit/course/InductiveSets2.pdf

14. Relevance Of The Axiom Of Choice
It s not as simple, aesthetically pleasing, and intuitive as the other axioms. of reals, and the BanachTarski Paradox (see the next section of the sci.math FAQ
http://db.uwaterloo.ca/~alopez-o/math-faq/node69.html
Next: Cutting a sphere into Up: The Axiom of Choice Previous: The Axiom of Choice
##### Relevance of the Axiom of Choice
THE AXIOM OF CHOICE There are many equivalent statements of the Axiom of Choice. The following version gave rise to its name: For any set X there is a function f , with domain , so that f x ) is a member of x for every nonempty x in X Such an f is called a choice function" on X . [Note that means X with the empty set removed. Also note that in Zermelo-Fraenkel set theory all mathematical objects are sets so each member of X is itself a set.] The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate. The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid's five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid's Parallel Postulate:
• Can it be derived from the other axioms?
• Is it consistent with the other axioms?
• Should we accept it as an axiom?
•  15. MATH 4023, Fall 2002, Homework (Group Axioms And Examples ) math 4023, Fall 2002, Homework (Group axioms and Examples )( 1) Problems from the text Chapter III, 1.1, 2.4 (replace monoidwith group), 2.6, 2.7, 2.8, 2.11, 2.13, 2.15.( If it is not a group, specify the rst in the sequence of. axioms that is not satised.http://www.math.lsu.edu/~chari/4023GroupHomework.pdf

16. Definition Of Set - WordIQ Dictionary & Encyclopedia
class (in the case of the von NeumannBernays-GÃ¶del axioms), and we if and only if it has no infinite descending membership sequence Â· Â· Â· math \in x2
http://www.wordiq.com/definition/Set
Encyclopedia Dictionary Thesaurus The Web eBooks loadkeyword("Set");
##### Set
Encyclopedia Definition: Set
In mathematics , a set is a collection of elements such that two sets are equal if, and only if, every element of one is also an element of the other. It does not matter in what order, or how many times, the elements are listed in the collection. By contrast, a collection of elements in which multiplicity but not order is relevant is called a multiset . Other related concepts are described below. If a set has n elements, where n is a natural number (possibly 0), then the set is said to be a finite set with cardinality n ; otherwise it is said to be an infinite set . Since all empty sets ( i.e. , sets with no elements) are equal to each other, it is permissible to speak of a set with no elements as the empty set For a discussion of the properties and axioms concerning the construction of sets, see the articles on naÃ¯ve set theory and axiomatic set theory . Here we give only a brief overview of the concept. Table of contents showTocToggle("show","hide")

 17. Math 315 Review Homework 1 1. Dene Field Axioms, Positivity Axioms math 315 Review Homework 11. Dene Field axioms, Positivity axioms and Completeness Axiom.2. Prove, directly from the axioms above, the following properties of realnumbers (i) ifhttp://math.gmu.edu/~tlim/315rhw1.pdf

18. Math 302 The Axioms
math 302 The axioms. An axiom means A proposition that commends itself to general acceptance; a well established or universallyconceded principle (OED2).
http://www.math.uiuc.edu/~stolman/m302/handouts/axioms.html
##### Math 302 The Axioms
An axiom means "A proposition that commends itself to general acceptance; a well established or universally-conceded principle..." (OED2). Often, one assumes the following statements are true. However, they are not true on every space. Therefore, we will check if each statement is true on each space.
• The "incidence axiom" There is at least one straight line between any two points. There is at most one straight line between two points. The "ruler axiom" You can travel an infinite distance along a straight line in either direction. As you travel along a straight line, you never pass over the same point twice. The "protractor axiom" There is at least one straight line through any point in any direction. There is at most one straight line through any point in any direction. The "half-plane" axiom If you cut the surface along a straight line, you get two pieces. Every straight line segment that connects two points on one of the pieces is contained entirely in that piece. The "mirror axiom" There is a global reflection through every straight line.
• 19. Math Forum: Ask Dr. Math FAQ: Dividing By Zero
Follow the links to read the full answers in the Dr. math archives is not so much physically impossible as it is in violation of mathematical axioms. You see
http://mathforum.org/dr.math/faq/faq.divideby0.html
Dividing by
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Classic Problems Formulas Search Dr. Math ... Dr. Math Home
##### Why can't you divide by 0? Why is 0/0 "indeterminate" and 1/0 "undefined"? Why is dividing by zero "illegal"?
Here, in their own words, are some explanations by our 'math doctors'. Follow the links to read the full answers in the Dr. Math archives.
Division by zero
Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related. 12 divided by 6 is 2 because 6 times 2 is 12 12 divided by is x would mean that times x = 12 But no value would work for x because times any number is 0. So division by zero doesn't work. - Doctor Robert
My teacher says you can't divide a number by zero. Why?
Let's look at some examples of dividing other numbers. 10/2 = 5 This means that if you had ten blocks, you could separate them into five groups of two. 9/3 = 3 This means that if you had nine blocks, you could separate them into three groups of three. 5/1 = 5 Five blocks could be separated into five groups of one. 5/0 = ? Into how many groups of zero could you separate five blocks? It doesn't matter how many groups of zero you have, because they would never add up to five since

20. Math Forum - Ask Dr. Math
That is how we think of math we choose some set of axioms (or postulates, which are the same thing) and definitions as our starting point, the things we are
http://mathforum.org/library/drmath/view/64481.html

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##### Flavors of Facts
Date: 10/03/2003 at 13:58:14 From: Karen Subject: Is 1 + 1 = 2 an actual fact? Is it a fact that 1 + 1 = 2? I have seen your proof using the Peano postulate. Is the postulate a hypothesis which is unproven, or is it proven, i.e., a fact? For example, 1 + 1 = 10 in base 2. So is the value of 1 + 1 open to interpretation? I think I find some of the terminology confusing, e.g., what do we really mean by the terms 'fact', 'premise', 'assumption', 'axiom', 'postulate', and so on? http://mathforum.org/library/drmath/view/62560.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 10/03/2003 at 17:45:37 From: Karen Subject: Thank you (is 1 + 1 = 2 an actual fact) Dear Dr. Math, Thanks for that excellent explanation. It has made things much clearer in my mind. In your explanation you make reference to "premises" and "assumptions". Are these the same thing, or are there subtle differences between the two? Regards, Karen http://mathforum.org/dr.math/

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