21. Probability Axioms - Wikipedia, The Free Encyclopedia respect to a universe or sample space math S /math of all possible elementary events in such a way that math P /math must satisfy the Kolmogorov axioms. http://www.phatnav.com/wiki/wiki.phtml?title=Probability_axioms

22. 03: Mathematical Logic And Foundations of nonstandard models for certain sets of axioms leads to nonstandard analysis of those axioms. MR82c03006; Smorynski, C. What s new in logic? math. http://www.math.niu.edu/~rusin/known-math/index/03-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 03: Mathematical logic and foundations Introduction Mathematical Logic is the study of the processes used in mathematical deduction. The subject has origins in philosophy, and indeed it is only by nonmathematical argument that one can show the usual rules for inference and deduction (law of excluded middle; cut rule; etc.) are valid. It is also a legacy from philosophy that we can distinguish semantic reasoning ("what is true?") from syntactic reasoning ("what can be shown?"). The first leads to Model Theory, the second, to Proof Theory. Students encounter elementary (sentential) logic early in their mathematical training. This includes techniques using truth tables, symbolic logic with only "and", "or", and "not" in the language, and various equivalences among methods of proof (e.g. proof by contradiction is a proof of the contrapositive). This material includes somewhat deeper results such as the existence of disjunctive normal forms for statements. Also fairly straightforward is elementary first-order logic, which adds quantifiers ("for all" and "there exists") to the language. The corresponding normal form is prenex normal form. In second-order logic, the quantifiers are allowed to apply to relations and functions to subsets as well as elements of a set. (For example, the well-ordering axiom of the integers is a second-order statement). So how can we characterize the set of theorems for the theory? The theorems are defined in a purely procedural way, yet they should be related to those statements which are (semantically) "true", that is, statements which are valid in every model of those axioms. With a suitable (and reasonably natural) set of rules of inference, the two notions coincide for any theory in first-order logic: the Soundness Theorem assures that what is provable is true, and the Completeness Theorem assures that what is true is provable. It follows that the set of true first-order statements is effectively enumerable, and decidable: one can deduce in a finite number of steps whether or not such a statement follows from the axioms. So, for example, one could make a countable list of all statements which are true for all groups.

23. 54: General Topology What is (are) topological dimension(s)? From the sci.math FAQ How can you chop Which settheory axioms are needed to prove there is no Banach-Tarski paradox http://www.math.niu.edu/~rusin/known-math/index/54-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 54: General topology Introduction More formally, a topological space is a set X on which we have a topology a collection of subsets of X which we call the "open" subsets of X. The only requirements are that both X itself and the empty subset must be among the open sets, that all unions of open sets are open, and that the intersection of two open sets be open. This definition is arranged to meet the intent of the opening paragraph. However, stated in this generality, topological spaces can be quite bizarre; for example, in most other disciplines of mathematics, the only topologies on finite sets are the discrete topologies (all subsets are open), but the definition permits many others. Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application. For example, a single point need not be a closed set in a topology. Does this seem "inappropriate"? Then perhaps you are envisioning a special kind of topological space, say a a metric space. This alone still need not imply the space looks enough like the shapes you may have seen in a textbook; if you really prefer to understand those shapes, you need to add the axioms of a manifold, perhaps. Many such levels of generality are possible.

24. Online Encyclopedia - Probability Axioms to a universe or sample space math \Omega /math of all possible elementary events in such a way that math P /math must satisfy the Kolmogorov axioms. http://www.yourencyclopedia.net/Probability_axioms.html

Encyclopedia Entry for Probability axioms Dictionary Definition of Probability axioms The probability event sample space elementary events Alternatively, a probability can be interpreted as a measure on a sigma-algebra of subsets of the sample space, those subsets being the events, such that the measure of the whole set equals 1. This property is important, since it gives rise to the natural concept of conditional probability independent Table of contents showTocToggle("show","hide") 1 Kolmogorov axioms 1.1 First axiom 1.2 Second axiom 1.3 Third axiom ... 2 Lemmas in probability Kolmogorov axioms First axiom That is, the probability of an event set is represented by a real number between and 1. Second axiom That is, the probability that some elementary event in the entire sample set will occur is 1, or certainty. More specifically, there are no elementary events outside the sample set. This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample set, then the probability of any subset cannot be defined either. Third axiom Any countable These axioms are known as the Kolmogorov axioms , after Andrey Kolmogorov who developed them.

25. Sci.math FAQ: The Continuum Hypothesis 1559 GMT ReplyTo alopez-o@neumann.uwaterloo.ca Archive-Name sci-math-faq/AC problem in set theory will be to lead to the discovery of new axioms which will http://www.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp/

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: The Continuum Hypothesis Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 Rate this FAQ N/A Worst Weak OK Good Great Current Top-Rated FAQs Are you an expert in some area? Share your knowledge and earn expert points by giving answers or rating people's questions and answers! This section of FAQS.ORG is not sanctioned in any way by FAQ authors or maintainers. Questions awaiting answers: 17142 questions related to FAQs 2985 general questions 8012 answered questions Usenet FAQs Search ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 05 2004 @ 00:23 AM

26. Godel And Godel's Theorem: Math of math, in terms of TNT, and in terms of Gödelized TNT. Zero equals zero is true. The string 0=0 is a valid TNT theorem (ie can be derived from axioms). http://www.ncsu.edu/felder-public/kenny/papers/godel.html

by Kenny Felder In the nineteenth and early twentieth centuries, one of the big mathematical goals was to reduce all of number theory to a formal axiomatic system. Like Euclid's Geometry, such a system would start off with a few simple axioms that are almost indisputable, and would provide a mechanical way of deriving theorems from those axioms. It was a very lofty goal. The idea was that this system would represent every statement you could possibly make about natural numbers. So if you made the statement "every even number greater than two is the sum of two primes," you would be able to prove strictly and mechanically, from the axioms, that it is either true or false. For real, die-hard mathematicians, the words "true" and "false" would become shorthand for "provable" or "disprovable" within the system. Russell and Whitehead's Principia Mathematica was the most famous attempt to find such a system, and seemed for a while to be the pinnacle of mathematical rigor. there is always a statement about natural numbers which is true, but which cannot be proven in the system. In other words, mathematics will always have a little fuzziness around the edges: it will never be the rigorous unshakable system that mathematicians dreamed of for millennia.

27. Traditional K-12 Math Education math skills. math facts consist of undefined terms, definitions, axioms (fundamental assumptions), and theorems. For example, the http://www.wgquirk.com/Genmath.html

Traditional K-12 Math Education Chapter 1 of The Truth About the NCTM Standards By William G. Quirk Knowledge Transmission Why Learn K-12 Math? How is K-12 Math Traditionally Learned? ... How is K-12 Math Traditionally Taught? Knowledge Transmission This chapter outlines the traditional "knowledge transmission" philosophy of K-12 math education. The fundamental assumptions are: Math is a man-made abstraction that only exists in the human mind or in written form. There is an established body of math knowledge that different people can understand in the same correct way. There is a stable foundational "K-12 math subset" that can be understood by different K-12 students in the same correct way. K-12 math teachers can lead K-12 students to a correct understanding of K-12 math. Why is K-12 Math Learned? The traditional reasons are: For the practical math-related needs of daily life. To prepare for occupations that use math. To develop the power of the mind to think logically and abstractly. To experience the step-by-step process of building a remembered knowledge base, relative to a structured knowledge domain.

28. Homage To Kurt Godel. were hard at work trying to prove consistency and completeness of the ZermeloFränkel formalism for set theory and logic, together with Peano s axioms for the http://www.chaos.org.uk/~eddy/math/Godel.html

The Berry Paradox (a cleaner variant on the `smallest non-interesting number' folly). could not be both consistent and complete; and could not prove itself consistent without proving itself inconsistent. The crucial technical terms of the discussion: Peano's axioms provide a formal description of the process of counting. They can be constructed in any logical system capable of the variety of counting in which any number has a successor - so that there is no `last' number - and distinct numbers have distinct successors. Consistency (of which the petty variety is the hobgoblin of small minds) is that desirable property of a logical system which says that there are no statements which the system regards as both true and false. Completeness is the desirable property of a logical system which says that it can prove, one way or the other, any statement that it knows how to address. ie it cannot be proven either true or false; in particular that it cannot be proven true. But `that it cannot be proven true' is Consequently, any logical system which can make up its mind about its consistency can prove itself inconsistent (provided it can count -

29. Probability Axioms Definition Meaning Information Explanation to a universe or sample space math Omega /math of all possible elementary events in such a way that math P /math must satisfy the Kolmogorov axioms. http://www.free-definition.com/Probability-axioms.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term Probability axioms The probability event sample space elementary event Alternatively, a probability can be interpreted as a measure on a sigma-algebra of subsets of the sample space, those subsets being the events, such that the measure of the whole set equals 1. This property is important, since it gives rise to the natural concept of conditional probability independent Inhaltsverzeichnis 1 Kolmogorov axioms 1.1 First axiom 1.2 Second axiom 1.3 Third axiom ... 2 Lemmas in probability Kolmogorov axioms First axiom That is, the probability of an event set is represented by a real number between and 1. Second axiom That is, the probability that some elementary event in the entire sample set will occur is 1, or certainty. More specifically, there are no elementary events outside the sample set. This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample set, then the probability of any subset cannot be defined either. Third axiom These axioms are known as the Kolmogorov axioms , after Andrey Kolmogorov who developed them.

30. Axiom Schema Of Specification Definition Meaning Information Explanation In the von NeumannBernays-G¶del axioms of set theory, a distinction is made between math exist D, forall C, C isin D harr left(Pleft(Cright) and exist E http://www.free-definition.com/Axiom-schema-of-specification.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term Axiom schema of specification In axiomatic set theory and the branches of logic mathematics , and computer science that use it, the axiom schema of specification , or axiom schema of separation , or axiom schema of restricted comprehension , is a schema of axiom s in Zermelo-Fraenkel set theory . It is also called the axiom schema of comprehension , although that term is also used for unrestricted comprehension, discussed below. Suppose P is any predicate in one variable that doesn't use the symbol B . Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: or in words: Given any set A , there is a set B such that, given any set C C is a member of B if and only if C is a member of A and P holds for C Note that there is one axiom for every such predicate P ; thus, this is an axiom schema. To understand this axiom schema, note that the set B must be a subset of A . Thus, what the axiom schema is really saying is that, given a set A and a predicate P , we can find a subset B of A whose members are precisely the members of A that satisfy P . We can use the axiom of extensionality to show that this set B is unique. We denote the set

31. MATH 3181-001 Fundamental Concepts Of Geometry math 3181 001 FALL 1996. axioms and Definitions for Plane Geometry. The Euclidean plane will be denoted by E 2 and the hyperbolic http://www.math.uncc.edu/~droyster/courses/fall96/math3181/Axioms.html

MATH 3181 - 001 FALL 1996 Axioms and Definitions for Plane Geometry The Euclidean plane will be denoted by E and the hyperbolic plane will be H . If we are referring to either of them, without specificity, we will use the notation P The plane is a set of points (undefined terms) and a line is a subset of the plane subject to the axioms that follow. If is a line and A is a point in that subset, we say that A lies on or passes through A Axiom 1: If A and B are distinct points, then there is one and only one line that passes through them both. Axiom 2: There is a function defined for all pairs of points P,Q such that and if and only if P=Q. This function satisfies the following properties: for all triples of points A B ,and C This axiom makes the plane into a metric space and is called the distance between P and Q Axiom 3: For each line , there is a one-to-one function, , such that if A and B are any points on AB x(A) - x(B) This mapping is not unique because we can use this same mapping and add a constant. On the other hand, it does say that each of our lines is isometric to the set of real numbers and, hence, are all isometric to one another.

32. Axioms Of Continuity It is addressed by the following principle, which is again a corollary of Dedekind s Axiom. Circular Continuity Principle . droyster@math.uncc.edu. http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node31.html

Next: Neutral Geometry Up: Neutral and Non-Euclidean Geometries Previous: Congruence Theorems Axioms of Continuity These axioms are the axioms which give us our correspondence between the real line and a Euclidean line. These are necessary to guarantee that our Euclidean plane is complete . The first axiom gives us some information about the relative sizes of segments as compared one to another. ARCHIMEDES' AXIOM. If AB and CD are any segments, then there is a number n such that if segment CD is laid off n times on the ray emanating from A , then a point E is reached where and B is between A and E This is derived from the Archimedean Axiom in the real number system. This should not be surprising, for we wish to have a one-to-one correspondence between each euclidean line and the set of real numbers . In the real line the Archimedean Postulate takes on the flavor: Archimedean Postulate: Let M and e be any two positive numbers. Then there is a positive integer n such that The main point for geometry is that if you choose any segment to be of unit length, then every segment has finite length with respect to this measure. Nothing can be too big

33. Godel's Theorems Quine s set theory would be just an historical footnote except for a longstanding open problem Does the consistency of Zermelo s axioms imply the consistency http://www.math.hawaii.edu/~dale/godel/godel.html

Godel's Incompleteness Theorem By Dale Myers Cantor's Uncountability Theorem Richard's Paradox The Halting Problem ... Godel's Second Incompleteness Theorem Diagonalization arguments are clever but simple. Particular instances though have profound consequences. We'll start with Cantor's uncountability theorem and end with Godel's incompleteness theorems on truth and provability. In the following, a sequence is an infinite sequence of 0's and 1's. Such a sequence is a function f Thus 10101010... is the function f with f f f A sequence f is the characteristic function i f i If X has characteristic function f i ), its complement has characteristic function 1 - f i Cantor's Uncountability Theorem. There are uncountably many infinite sequences of 0's and 1's. Proof . Suppose not. Let f f f , ... be a list of all sequences. Let f be the complement of the diagonal sequence f i i Thus f i f i i For each i f differs from f i at i Thus f f f f This contradicts the assumption that the list contained all sequences.

34. Topos describes ordinary set theory in topostheoretic terms, making it clear which axioms will be is a philosopher by profession, but this is very much a math book. http://math.ucr.edu/home/baez/topos.html

Topos Theory in a Nutshell John Baez January 27, 2004 Okay, you wanna know what a topos is? First I'll give you a hand-wavy vague explanation, then an actual definition, then a few consequences of this definition, and then some examples. I'll warn you: despite Chris Isham's work applying topos theory to the interpretation of quantum mechanics, and Anders Kock and Bill Lawvere's work applying it to differential geometry and mechanics, topos theory hasn't really caught on among physicists yet. Thus, the main reason to learn about it is not to quickly solve some specific physics problems, but to broaden our horizons and break out of the box that traditional mathematics, based on set theory, imposes on our thinking. 1. Hand-Wavy Vague Explanation Around 1963, Lawvere decided to figure out new foundations for mathematics, based on category theory. His idea was to figure out what was so great about sets, strictly from the category-theoretic point of view. This is an interesting project, since category theory is all about objects and morphisms. For the category of sets, this means SETS and FUNCTIONS. Of course, the usual axioms for set theory are all about SETS and MEMBERSHIP. Thus analyzing set theory from the category-theoretic viewpoint forces a radical change of viewpoint, which downplays membership and emphasizes functions. Even earlier, this same change of viewpoint was also becoming important in algebraic geometry, thanks to the work of Grothendieck on the Weil conjectures. So topos theory can be thought of as a merger of ideas from geometry and logic - hence the title of this book, which is an excellent introduction to topos theory, though not the easiest one:

35. Spin.stat From baez@math.ucr.edu (john baez) Newsgroups sci.physics,sci.math Subject Re Spin The key axioms for quantum fields that one must assume in order to deduce http://math.ucr.edu/home/baez/spin.stat.html

Spin, Statistics, CPT and All That Jazz John Baez April 6, 1997 This is a little grab-bag of proofs of the spin-statistic relations, with some stuff about CPT thrown in. First for two fun topological "proofs," and then a more rigorous highbrow one! For more details, check out R. F. Streater and A. S. Wightman's PCT, Spin and Statistics, and All That, reprinted by Addison-Wesley, New York, 1989. From: baez@math.ucr.edu (john baez) Newsgroups: sci.physics,sci.math Subject: Re: Spin/Statistics Heuristic Argument Date: 31 Jan 1995 18:58:22 -0800 Thanks for the reference. Jim Dolan and I have been thinking about this kind of thing lately. The standard proof involves lots of assumptions, but there should be a topological - actually category-theoretic - essence to it, and this heuristic argument is probably pretty close to the theorem we're looking for. Actually I already know another heuristic argument, but this one might be fundamentally a bit different. >The "proof" is topological, in that if you rotate a thing (particle) >with strings all over it, the strings are all twisted after 360 >degrees of rotation, but after 720 degrees the strings can be >untangled without moving the object. Similarly, if two things >(particles) connected by lots of strings are interchanged, the strings >are left twisted up exactly as if one particle had been rotated by 360 >degrees. So the conclusion is that interchanging two particles is >topologically indistinguishable from a rotation of one particle by 360 >degrees - a particle which changes sign after a rotation will be >antisymmetric wrt pairwise interchange.

36. Automated Reasoning 1898 Huntington s axioms. Edward V. Huntington, Sets of independent postulates for the algebra of logic, Trans. Am. math. Soc., vol. http://www.math.princeton.edu/~nelson/ar.html

Automated Reasoning This is material related to MAT 504, Topics in Logic, for the Spring term of 1997 (MW 1:30-3:00, Fine 214), on automated reasoning. These are not full lecture notes for the course, and sometimes there is only a list of topics presented at the blackboard or computer terminal. Here is the relevant directory index , and this is my home page Blurb After the media hype about the computer solution of the Robbins problem is discounted, what remains? This question can be answered only by looking into what was done. In my opinion, it is a significant achievement. I am skeptical about the role that automated reasoning can play in mathematics in the next few centuries, but there are interesting and non-trivial results. They lead us into an odd mixture of computer science, logic, universal algebra, pragmatic experimentation, and (so help me) elliptic curves. The goals of this course are to describe the methodology of the field, to have fun playing with the programs Otter, Mace, and EQP that led to the solution of the Robbins problem, to look at some other programs that have been developed, to explore some of the corners of mathematics in which automatic reasoning has proved useful, to help anyone seriously wishing to apply these tools to problems of interest, and to try to estimate the promises and limitations of automatic reasoning. Although I have worked on a proof-checking program, which I hope to have in shape to publish this year, I am not a member of the AR (automated reasoning) community and have no expertise in the field. There may well be some boners here and if so I hope that more knowledgeable people will set me (

37. 1998: A Good Year For Math? prizewinner John Nash, and you have a media environment in which math is floating not), one way to axiomatize boolean algebras is by the following four axioms http://www.maa.org/devlin/devlin_12_98.html

Devlin's Angle December 1998 1998: A Good Year for Math? Was 1998 a good year for mathematics or a poor one? It depends on the yardstick you use to measure performance. In terms of media attention, it has probably been the best year ever (see my November column, "Math Becomes Way Cool"). But if you are looking for major new results, it has been a relatively poor year. At about this time each year for the past four years, the American Mathematical Society has published an attractive (full color, high gloss) pamphlet titled What's Happening in the Mathematical Sciences. Written by mathematics writer Barry Cipra, WHIMS sets out to do what its title suggests: It provides accounts of the most significant developments in mathematics over the previous twelve months, written at an accessibility level roughly the same as Scientific American. Volume 4 of WHIMS appeared recently, and it's as good as the previous three issues. Barry Cipra is to be praised for his excellent mathematical writing, Paul Zorn has done a magnificent job as editor, and the AMS does the mathematical community a great service by publishing this annual summary. One thing that struck me as I read this latest issue, however, is that all the developments described occurred in previous years. Now, don't get me wrong. I am not drawing any conclusions from this observation. For one thing, the acquisition of human knowledge proceeds at its own, often erratic pace. For another, there is more to mathematics than major breakthroughs that hit the front pages of daily newspapers.

38. Lecture Notes 2 - Math 3210 axioms for the Finite Geometry of Pappus. There exists at least one line. axioms for the Finite Geometry of Desargues. There exists at least one point. http://www-math.cudenver.edu/~wcherowi/courses/m3210/hg3lc2.html

Lecture Notes 2 Pappus of Alexandria (340 A.D.) Pappus' Theorem: If points A,B and C are on one line and A', B' and C' are on another line then the points of intersection of the lines AC' and CA', AB' and BA', and BC' and CB' lie on a common line called the Pappus line of the configuration. Axioms for the Finite Geometry of Pappus There exists at least one line. Every line has exactly three points. Not all lines are on the same point. [N.B. Change from the text] If a point is not on a given line, then there exists exactly one line on the point that is parallel to the given line. If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'. With the exception in Axiom 5, if P and Q are distinct points, then exactly one line contains both of them. Theorem 1.10 Each point in the geometry of Pappus lies on exactly three lines. Pf . Let X be any point. By corrected axiom 3, there is a line not containing X. This line contains points A,B,C [Axiom 2]. X lies on lines meeting two of these points, say B and C [Axiom 5]. There is exactly one line through X parallel to BC [Axiom 4]. There can be no other line through X since by Axiom 4 it would have to meet BC at a point other than A, B or C [Axioms 6 and 5], and this would contradict Axiom 2. Pappus geometry has 9 points and 9 lines.

39. Lecture Notes 6 - Math 3210 The term is usually applied only to the special geometries that are obtained by negating the parallel postulate but keeping the other axioms of Euclidean http://www-math.cudenver.edu/~wcherowi/courses/m3210/hg3lc6.html

Lecture Notes 6 Non-Euclidean Geometry is not not Euclidean Geometry. The term is usually applied only to the special geometries that are obtained by negating the parallel postulate but keeping the other axioms of Euclidean Geometry. History of the Parallel Postulate. Saccheri Euclid Freed of Every Flaw " (1733, published posthumously) The first serious attempt to prove Euclid's parallel postulate by contradiction. This Jesuit priest succeeded in proving a number of interesting results in hyperbolic geometry, but reached a flawed conclusion at the end of the work. Lambert Theory of Parallels " (also published posthumously) Similar in nature to Saccheri's work, and probably influenced by it. However, Lambert was astute enough to realize that he had not proved the parallel postulate. He did not publish this work himself. Nikolai Ivanovich Lobachewsky On the Principles of Geometry The first published account of hyperbolic geometry, in Russian. Lobachewsky developed his ideas from an analytical (trigonometric) viewpoint. Johann ('Janos') Bolyai Appendix exhibiting the absolute science of space: independent of the truth or falsity of Euclid's Axiom XI (by no means previously decided) " in Wolfgang Bolyai's

40. Math Teacher Evaluation of Learning (SOLs), the following rating scale for math Teachers at GEOMETRY, Creates consistent sets of axioms, Proves original theorems, Accepts axioms, Proves http://www.pen.k12.va.us/Div/Winchester/jhhs/math/humor/eval.html

Math Teacher Evaluation In accordance with the new emphasis on accountability and the Standards of Learning (S.O.L.s), the following rating scale for Math Teachers at Handley High School has been devised: Rating Scale Far Exceeds Job Requirements Exceeds Job Requirements Meets Job Requirements Needs Some Improvement Does Not Meet Minimum Requirements QUALITY Leaps tall buildings with a single bound Must take running start to leap over tall buildings Can leap over short buildings only Crashes into buildings when attempting to jump over them Cannot recognize buildings at all TIMELINESS Is faster than a speeding bullet Is as fast as a speeding bullet Not quite as fast as a speeding bullet Would you believe a slow bullet? Wounds self with bullet when attempting to shoot INITIATIVE Is stronger than a locomotive Is stronger than a bull elephant Is stronger than a bull Shoots the bull Smells like a bull ADAPTABILITY Walks on water consistently Walks on water in emergencies Washes with water Drinks water Passes water in emergencies COMMUNICATION

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