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Do left sided identity and right sided inverse suffice to recover both sided axioms in group theory?
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∙ 16y ago
No. For a counterexample, define a*b=b for all a and b in the group. Then we can pick any e to be the left identity of all the elements. Similarly, any b has the right inverse e because b*e=e. However, (if there is more than one element), this doesn't satisfy the conditions on a group because there is no single (two-sided) identity element. If a*x=a and b*x=b, then x=a and x=b, which obviously can't hold in the general case.
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∙ 16y ago
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Merits of quantum theory over classical theory?
Unlike other physical theories, quantum mechanics was the invention of not only one or two scientists. Planck, Einstein, Bohr, Heisenberg, Born, Jordan, Pauli, Fermi, Schrodinger, Dirac, de Broglie, Bose are the scientists that made notable contributions to the invention of quantum theory. The axioms of quantum mechanics provide a consistent framework in which it is once again possible to predict the results of experiment, at least statistically.Its fundamental features are that a property does not exist unless it is measured, and that indeterminacy is a fundamental property of the universe. The main merit of QM is that its predictions -- such as that for the two slit experiment -- perfectly match the results, while classical mechanics fails to do so. For a scientist, nothing else much matters.
Why don't all theories become laws?
Scientific theories and laws technically are never true. They are models that scientists create in order to predict outcomes. Since nature has given us no essential axioms everything scientists have done is just a model. For example: When maxwell originally developed his theory he used a set of cogs and wheels to explain electromagnetism. He later realized these were not necessary and dropped them from his formulation. This changes the essence of his theory but both ways delivered the same predictions given a set of data points. Similarly, before Einstein, newtons laws were considered infallable however later we realized that newton's laws only work in certain realms. Even Einstein didn't have the whole story. Dirac later applied his relativistic theories to the quantum field. Currently dirac's theory has the largest realm of applicablity (although it is very difficult to use). However even dirac's theory cannot explain all realms. Therefore theories and laws, since they are not essential to nature and are just models and can be replaced.
What is the interactional model of communication?
Interactional ViewFamily Communication- Family is a system (cybernetic tradition)- Watzlawick: individuals must be understood within a family system (psychiatry)- Family (system) supersedes the individual regarding communication within the family- Relationships: complex functions- Each family: unique game; unique realityAxioms of Interpersonal Communication- Homeostasis: Status quo; strong compulsion to maintain- Resistance to change: Destructive.Must understand axioms: "grammar" or rules of "game"- One cannot not communicateOne cannot not influence- Communication = content + relationshipLatter determines the formerMeta-communication: Communication about communication- Early definition: Meta-communication is any relational communication- Meta-communication dominates when family/relationship in trouble- "Sick" relationships only get better when members are willing to acknowledge meta-communication- Relationship depend on "punctuation"Where does one mark the beginning of an interaction?Problems arise when individual consistently sees himself as reacting- All communication is either symmetrical or complementaryControl, status, power are bedrock assumptions of the interactional viewSymmetrical interchanges: predicated on equal power- Complimentary interchanges: predicated on differential power- Healthy relationships: both kinds of communicationRogers: "reciprocal complementarity" - shared dominance - correlates with relational satisfaction- Must assess an exchange of at least two messages to determine nature of exchange- Coding systemo One-up communicationo One- downo One- across- Rogers: Flexibility highly significant predictor of relational satisfaction- Family systems are highly resistant to change- Double binds: competing/contradictory demands on members of the system- Paradox: High status member demands that low-status member behave as if relationship is symmetricalo Example: "second shift"- Reframing: changes the games via the rulesAltering punctuationMust analyze from outside the systemAdopt a new frameUsually requires outside assistanceCritiqueModifications:- Not all nonverbal behavior is communication (i.e, you cannot not communicate). Difference between information and communication- The term "meta-communication" should be reserved for explicit communication about communication, not all relational communicationEquifinality: Outcomes could be cause by any number of factors and/or combinations of them
What was one of sir Isaac newtons contribution to physics?
It is all about Newton's achievements including his theory of universal gravitation, his famous laws of motion, his study of light, and his studies on calculus. Sir Isaac Newton : Contributions One of the most important scientists of all time, Isaac Newton, made many discoveries and theories that have changed the world. His studies in physics have influenced modern physics greatly with his laws of motion, his study of light, and his law of gravitational motion. Newton also created one of the most important scientific books of all time, the Principia, widely regarded as one of the most influential works on physics of all times. Newton has been one of the most influential and important people throughout history with his theories and his studies. Isaac Newton, one of the greatest English scientists and mathematicians, was born in Woolsthorpe, Lincolnshire. He was born on December 25 1642, and was born "posthumously and prematurely and barely hung onto life" and "had an ill-starred youth"(Asimov103). Isaac Newton grew up from a family of farmers and he became very prosperous because of it. He never knew his father, who was also named Isaac Newton, who died in October 1642, three months before his son was born. Although Newton's father owned a great deal of property and animals which made him a wealthy man, he was uneducated and could not sign his own name (Connor). His mother, marrying again three years later, left the child with his grandparents. Isaac Newton's life can be split up into 3 different periods. The first period is his boyhood days from 1643 up to his appointment to a chair in 1669. The second period from 1669 to 1687 was where he did most of his work and was a professor at Cambridge. The third period was when Newton was a government official and had little interest in mathematical interest. (Connor). At school, Newton was interested in constructing mechanical devices. He showed no signs of unusual brightness. "He was slow in his studies until his late teens. He was even taken out of his studies in the late 1650s" (Asimov 103). His uncle who attended Cambridge College detected a scholar in Newton, and he urged him to go to Cambridge. In 1660, Isaac attended Cambridge, and in 1665 he graduated. Newton had to leave Cambridge because of the plague, and it was during this time that Newton developed most of his significant discoveries. Because Newton was very reclusive, Newton did not, however, publish his results. (Weisstein). Newton's four greatest achievements was the study of light, the invention of the binomial theorem, the discovery of calculus, and the theory of universal gravitation. Newton's prism experiments made him famous. "Newton, at 27 became professor of mathematics at Cambridge. He was elected to the Royal Society in 1672, where he reported his experiments on light and color to the Society" (Asimov 103). Newton rarely went to bed till two or three o' clock, sometimes not till five or six. He used to employ six weeks in his laboratory till he finished his experiments. But Newton's fame also brought him some enemies. Hooke was Newton's main rival. He started this rivalry when Newton reached the Royal Society (Asimov, 106). Hooke accused Newton of stealing the idea of the inverse square law. "Hooke, a member of the Royal Society, was a prolific man when it came to scientific inventions and theories" (Muir 115). He attacked Newton for stealing his ideas and maintained a lifelong enmity, clearly founded on jealousy. Newton was one of the most well known scientists in his time, "as Newtonian science became increasingly accepted on the Continent" (Muir 117). Newton was "at the height of his creative power, he singled out 1665-1666 (spent largely in Lincolnshire because of plague in Cambridge) as the prime of my age for invention." During those couple of years, he prepared Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) also known as the Principia which has one of the biggest impacts on today's Physics (Hull). It is perhaps the "most powerful and influential scientific treatise known to man" ("Newton"). In his book he covered his discovery of calculus, theories for light and color, and planetary motion ("Newton"). Book I begins with eight definitions and three axioms. These are later called Newton's Laws of Motion ("Newton"). In Book II, Newton treats the motion of bodies through restating mediums as well as the motion of fluids. He discredited Descartes system of planetary laws, saying that the vertices of Descartes could not be self-sustaining. Book II was not originally part of the original outline ("Newton"). Newton's Book III, also called "System of the World," stated the famous law of Universal gravitation. Newton used gravitational attraction to explain the motion of the planets and their moons. Newton's laws became the physical and intellectual foundation of the modern world view ("Newton"). The Principia is widely regarded as the most "important and influential works on physics of all time" with many of those theories influencing modern physics today (Weisstein). Newton's three laws of motion are described in Newton's first book of Principia. Newton's first law is applications for modern physics. Newton's First Law of Motion states that "a body remains at rest and a body in motion remains in motion at a constant velocity as long as outside forces are not involved." This law is also known as the Principle of Inertia. Newton's second law "defines a force in terms of mass and acceleration and this was the first clear distinction between the mass of the body and its weight." Newton's third law states that "for every action there is an equal and opposite reaction." This law is used today in modern rockets and is his most famous out of his laws. Newton was able to find out how the gravitational force between the earth and moon could be calculated with his three laws. (Asimov, 108). Newton's three laws can be characterized as the foundation of the theory of motion. These laws have revolutionized modern physics and are one of the most renowned theories in physics. One of Newton's greatest achievements was his theories of light. Newton's most famous experiments, experimentum crucis, demonstrated his theory of the composition of light. Newton let light pass through a prism and what was one beam of light, 3 came out of the prism ("Newton") Newton investigated further and discovered that light was made up of seven colors which bend and refract at different angles (Muir, 107). Newton thought that light rays moved in straight lines rather than a wavelike motion. His study in light greatly influenced the refracting telescopes making them stronger and clearer (Asimov, 107). Newton's study of light was the beginning of optics and the study of light (Muir, 107). Newton was greatly renowned for his study in light which he published his studies in his second book Optiks. According to a well-known story, many people believed that Newton saw a apple fall in his orchard and developed a theory that the same force governed the motion of the Moon and the apple. (Hull). Newton theorized that the rate of fall was proportional to the strength of the gravitational force and that this force fell off according to the square of the distance to Earth. "A law of attraction held between any two bodies in the universe, so that his equation became the law of universal gravitation. It explained all of Kepler's Laws and explained planetary motion as it is today" (Asimov, 109). The Law of Universal Gravitation not only altered men's mind about divine bodies, but changed men's minds about human minds (Muir, 112). Newton's achievement in the law of universal gravitation is well known, and has revolutionized modern physics. Newton is also given credit to have huge achievements in mathematics as well as physics. Newton invented the binomial theorem and calculus while he was studying the infinite series. Newton and Leibniz developed the calculus independently and at about the same time. This sparked a rivalry between them on who developed the theory first. Newton is given credit to have discovered it. But at the same time Leibniz published his work before Newton, so Leibniz should have gotten credit for it (Muir, 107). Newton's research in mathematics has made him not only one of the greatest scientists of all time, but has also made him one of the greatest mathematicians. Throughout history, there has been no other more influential scientist than Isaac Newton. Newton's study in physics and mathematics put him with many of the greatest scientists. His three laws of motion, his study of the nature of light, and his law of universal gravitation are some of his greatest works. Newton has been thought of for the last 300 years, the founding exemplar of modern physical science. So it is therefore no exaggeration to see that Newton was and still is the single most important contributor to the development of modern science.
Show that the set of integers with respect to multiplication is not a group?
To be a group, the set of integers with multiplication has to satisfy certain axioms: - Associativity: for all integers x,y and z: x(yz) = (xy)z - Identity element: there exists some integer e such that for all integers x: ex=xe=x - Inverse elements: for every integer x, there exists an integer y such that xy=yx=e, where e is the identity element The associativity is satisfied and 1 is clearly the identity element, however no integer other than 1 has an inverse as in the integers xy = 1 implies x=y=1
What does the word paddock mean in math?
A paddock is a set that satisfies the 4 addition axioms, 4 multiplication axioms and the distributive law of multiplication and addition but instead of 0 not being equal to 1, 0 equals 1. Where 0 is the additive identity and 1 is the multiplicative identity. The only example that comes to mind is the set of just 0 (or 1, which in this case equals 0).
When was Peano axioms created?
Peano axioms was created in 1889.
When was Axioms - album - created?
Axioms - album - was created in 1999.
What are axioms in algebra called in geometry?
They are called axioms, not surprisingly!
Why does the fact 0 has no multiplicative inverse still mean R is still a field?
In the definition of a field it is only required of the non-zero numbers to have a multiplicative inverse. If we want 0 to have a multiplicative inverse, and still keep the other axioms we see (for example by the easy to prove result that a*0 = 0 for all a) that 0 = 1, now if that does not contradict the axioms defining a field (some definitions allows 0 = 1), then we still get for any number x in the field that x = 1*x = 0*x = 0, so we would get a very boring field consisting of only one element.
Axioms must be proved using data?
Axioms cannot be proved.
If a figure has 2 right angles are the angles congruent?
PostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-Petroz
Which are accepted without proof in a logical system?
axioms
What terms are accepted without proof in a logical system geometry?
Such terms are called axioms, or postulates.Exactly which terms are defined to be axioms depends on the specific system used.
What are the 4 fundamental laws in mathematics?
I don't know why there should be 4 laws (=axioms) specifically. In mathematics you can choose whatever system of axioms and laws and work your way with those. Even "logic" (propositional calculus) can be redefined in meaningful ways. the most commonly used system is Zermolo-Fraenkel+choice: http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms It has 9 axioms though, not 4. One might want to take into consideration the rules of "logic" as basic laws: http://en.wikipedia.org/wiki/Propositional_calculus Another common set of axioms that can be created inside the ZFC system is peano arithmetic: http://en.wikipedia.org/wiki/Peano_arithmetic I hope I understood your question. The short answer is "there is no such thing". I think the questioner may have meant the 5 fundmental laws in mathematics, also known as the axioms of arithmetic, these are as follows: A1 - for any such real numbers a and b, a+b=b+a, the commutative law A2 - for any such real numbers a,b and c, a+(b+c) = (a+b)+c, the associative law A3 - for any real number a there exists an identity, 0, such that, a+0 = a, the identity law A4 - for any real number a there exists a number -a such that a+(-a)=0, the inverse law A5 - for any real numbers a and b, there exists a real number c, such that a+b=c, the closure property. These 5 axioms, when combined with the axioms of multiplication and a bit of logic/analytical thinking, can build up every number field, and from there extend into differentiation, complex functions, statistics, finance, mechanics and virtually every area of mathematics.
Do axioms and postulates require proof?
No. Axioms and postulates are statements that we accept as true without proof.
