What is Definitions: Definition and 273 Discussions

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed.

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  1. Fredrik

    Definitions of topology and analysis

    Definitions of "topology" and "analysis" How do you define "topology" and "analysis"? I'm tempted to say that topology is the mathematics of...anything that involves limits. (Open and closed sets, continuous functions, etc...they can all be defined in terms of limits). But if that's an...
  2. Dembadon

    Defining Polygons with Precision: A Review of Basic Polygon Terminology

    I'm working through the following book: Principles of Mathematics, by Allendoerfer & Oakley. Since I haven't taken a proof-based course yet, and won't be able to until spring 2012 :grumpy:, I want to make sure that I'm not forming habits that will hurt me when I do. There are some answers that...
  3. P

    Wave Definitions - Crest and Trough

    Hi, I have a test on the definitions of some terms. I was wondering whether the crest is the highest point in a wave or any point above the undisturbed position. And, also, whether the trough is the lowest point or any point under the undisturbed position. Thanks, Peter G.
  4. N

    Two definitions of wave reflection coefficient

    The waves book by A.P.French gives two characterizations of the reflection coefficient for a 1-D traveling wave encountering an interface between two media. On one hand, he writes R = \frac{v_2 - v_1}{v_2 + v_1} where v_i are the wave speeds in the two media. Later on, he writes the reflection...
  5. O

    Equivalent definitions of Equivalent metrics.

    Two metrics 'd' and 'f' are said to be equivalent on a metric space X, if they have the same set of open sets. This is equivalent to saying every open ball with respect to d contains an open ball with respect to f (different radius) and vice versa. (As every open set is a union of open balls)...
  6. L

    Definitions and if and only if statements

    I'm trying to learn some analysis on my own, and as this is the first proof-based book I'm reading, I have a basic question about definitions I was hoping someone could help me with. For example, the book I'm reading says that: Given a subset of the real numbers A, b is an upper bound of A if...
  7. E

    Definitions of vector space and subspace

    I am using Axler's Linear Algebra Done Right as a text for independent study of linear algebra. Axler basically defined a vector space to be a set which has defined operations of addition and multiplication (and which comports with certain algebraic properties) and that contains an additive...
  8. B

    What Is the Difference Between Latent Heat of Fusion and Specific Melting Heat?

    Homework Statement I\'m just trying to understand a definition: So for the clausius clapeyron equation, it really matters which way we define L (i.e. whether it is positive or negative), so trying to understand, is \"the latent heat of fusion of ice\" the same as \"the specific melting...
  9. Rasalhague

    Understanding Probability Spaces: An Example Using Double Coin Tosses

    I'd like to test my understanding of some basic definitions using the example of a double coin toss. I think this would be formally modeled with the following structure: A1. The sample space, S = {(0,0),(0,1),(1,0),(1,1)}, whose elements (s1,s2) are called outcomes, where si = 1 means heads...
  10. M

    Are the definitions of vectors in Liner Algebra and physics contradictory?

    I'm really confused. I study physics, and in Relativity we deal a lot with vectors and tensors. I know an element x of Rn is a vector, because Rn is a vector space. That's fine. Now, in physics, we define a (contra-) vector xa (with 'a' an index running from 1 to n) as a n-tuple of quantities...
  11. M

    Are the definitions of vector in Linear Algebra and physics compatible?

    I'm really confused. I study physics, and in Relativity we deal a lot with vectors and tensors. I know an element x of Rn is a vector, because Rn is a vector space. That's fine. Now, in physics, we define a (contra-) vector xa (with 'a' an index running from 1 to n) as a n-tuple of...
  12. C

    Thermodynamics, confused with definitions

    Homework Statement dU = dQ + dW so dU = Cv dT... but only if dW is zero right? as dQ = CvdT so then the central equation is TdS = dU + PdV which is then TdS = CvdT + PdV according to my notes. surely PdV should be zero no?
  13. C

    How many definitions of holomorphic

    There are a lot of definitions but what is the quickest way to see if a function is holomorphic? apply the cauchy riemann equations seems too slow. I thought if it doesn't have a z_bar in it, then it's automatically holomorphic. so for ex. polynomials are always holomorphic. on the other...
  14. Fredrik

    Two definitions of locally compact

    I'm trying to understand the proof of (ii)\Rightarrow(i) of proposition A.6.2.(1) here. The theorem says that the given definition of "locally compact" is equivalent to a simpler one when the space is Hausdorff. I found the proof quite hard to follow. After a few hours of frustration I'm down to...
  15. S

    Calculus: Increasing/Decreasing, Critical Points & Inflection Points

    srry if this post is in the wrong section but i was wondering if there are actually precise and universally agreeable definitions of the following terms of calculus. Different textbooks even give contrary definitions. Any help is appreciated thanks. increasing/decreasing = strictly...
  16. A

    Equivalent definitions for the norm of a linear functional

    Can someone please explain why the following three definitions for the norm of a bounded linear functional are equivalent? \| f \| = \sup_{0 < \|x\| < 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{0 < \| x \| \leq 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{\| x \| = 1}...
  17. Rasalhague

    Defining a Flow: What's the Best Way?

    I'm wondering which of the following definitions of a flow is best. Is there one primary, rigorous, general definition of which the others are informal shorthands, or are the differences no more then superficial differences in convention? (1) Arnold, in Ordinary Differential Equations...
  18. Rasalhague

    Why is Arnold's extended phase space a strip rather than a rectangle?

    I'm reading Arnold: Ordinary Differential Equations, Chapter 1. In section 1.2, an integral curve was defined as the graph, in the extended phase space, \mathbb{R} \times M, of the motion \phi : \mathbb{R} \rightarrow M of a phase point in M. In 2.2, an integral curve is defined as the graph of...
  19. E

    Main branches of math definitions

    https://www.physicsforums.com/showthread.php?t=466170 I'm currently trying to get better at math. I've decided to do research on every facet of mathematics, starting at the source: what is a number? And going from there, getting progressively more advanced. Instead of just trying to...
  20. C

    Definitions of eigenstate, eigenvalue and eigenfunction?

    Homework Statement In quantum mechanics a physical observable is represented by an operator A. Define the terms eigenstate, eigenvalue and eigenfunction of a quantum mechanical operator. Homework Equations The Attempt at a Solution I think I know in that eq 'f' is the eigenfunction, and...
  21. Rasalhague

    Meaning of countable in definitions of sigma algebra

    Meaning of "countable" in definitions of sigma algebra In the third axiom defining a \sigma-algebra, (X,\Sigma), does countable mean (a) "finite or countably infinite", or does it mean (b) "countably infinite".
  22. K

    Differing definitions of an inner product

    Hey all, This might seem like a stupid question, and this might not be the correct forum, but hopefully someone can clarify it really easily. I often have seen two definitions of an inner product on a vector space. Firstly, it can be defined as a bilinear map on a \mathbb F-vector space V...
  23. H

    Definitions of greatest and least elements in terms of strict orderings

    Homework Statement State the definitions of greatest and least elements in terms of strict orderings. Homework Equations Let \leq be an ordering of A and < be a strict ordering on A, and let B \subseteq A. b \in B is the greatest element of B in the ordering \leq if, for every x \in...
  24. I

    What is Energy? - Unpacking Hard Definitions

    What is Energy? - Hard energy definitions Explain this definition: energy is the ability to impart vis viva (mv2) Is it a well definition? Here's a more complex definition: energy is that measure of the physical change of a system that is conserved as a result of temporal displacement...
  25. S

    Crystal-Bravis Lattice Definitions

    There is such thing as a orthorhombic body centered crystal lattice. I am wondering why this is the case see the image bellow, we can find a repeating pattern which has a smaller area. A unit cell - must be selected such that it has the highest symmetry and the smallest area, however i do...
  26. bcrowell

    Differing definitions of expansion, shear, and vorticity

    There is a discussion of expansion, shear, and vorticity in Wald (p. 217) and in Hawking and Ellis (p. 82). My motivation for comparing them was that although Wald's treatment is more concise, Wald doesn't define the expansion tensor, only the volume expansion. Wald starts off by restricting to...
  27. I

    What Are the Core Concepts of Model Theory?

    Hey I've been reading the basic definitions for model theory, and got a bit confused, maybe someone can help me? That's how I understood the definitions: An m-Type in a model M is a set of formulas (with m variables), such that it is finitely satisfiable An m-Type over A in M is a set of...
  28. stripes

    Analytical definitions vs intuitive (or perhaps first year ) definitions

    Analytical definitions vs intuitive (or perhaps "first year") definitions I just began my real analysis course in college and we were given an assignment; a bunch of mathematical terms for us to define. We are asked to define them using two textbooks, one, our first year calculus textbook, the...
  29. C

    Bragg Diffraction angle definitions

    I find problems on Bragg diffraction frustrating. I can't tell how the angles are defined, nor the "planes" in the crystal--they look arbitrary. Why can't I just draw a slash through the crystal at any angle I want and get diffraction off the angles I hit with the slash? Is it just that the...
  30. C

    Exploring Mass & Establishing Unit of Mass

    In physics, we use the concept of mass, and we use a unit of mass. It would appear that we must first define the concept of mass, to get to a position where we can establish a unit of mass. Mass not rigorously defined To my knowledge the concept of mass is not rigorously defined in Physics. We...
  31. G

    Mathematica [Mathematica] Saving definitions

    Hi, I'm trying to save (into a file) definitions of some variables which are of the form Subscript[A,1], Subscript[A,2],... where the subscript is used as an index for the variable. When using the Save command I obtain the error Save::sym: Argument A1 at position 2 is expected to be a...
  32. N

    I have an intuitive understanding of the definitions that will follow,

    I have an intuitive understanding of the definitions that will follow, but in my search to find specific definitions I only come up with vague explanations and contentious or subjective uses of those definitions. Please provide some definitions (mathematical) for the following list (and sources...
  33. S

    Series: Definitions and Properties problem

    Homework Statement [PLAIN]http://img528.imageshack.us/img528/7061/37557155.jpg Homework Equations The Attempt at a Solution i don't understand, what can possibly wrong with this :S I checked it so many times over and over but everything seems to be right.. i don't even have this feeling about...
  34. S

    Inner Product Definitions Galore?

    Hello, I thought I understood the Dot Product but Apparently Not! \overline{u} \ \cdot \ \overline{v} \ = (u_x \ \cdot \ v_x) ( \overline{i} \cdot \overline{i} ) \ + \ (u_y \ \cdot \ v_y) ( \overline{j} \cdot \overline{j} ) \ = \ | \overline{u} | | \overline{v} | cos \theta That is the...
  35. M

    Understanding Epsilon Delta Definitions for Limits: Functions that Satisfy Them

    Homework Statement Im trying to figure out what the difference is between the following two epsilon delta statements and the kinds of functions they satisfy: For all real numbers x and for all delta>0, there exists epsilon>0 such that |x|<delta implies |f(x)|<epsilon vs. there exists...
  36. DaTario

    Entanglement and Concurrence: asking for definitions

    Hi All I would like to know if one can present simple definitions for entanglement and concurrence as well as experimental forms to detect them. Sincerely DaTario
  37. T

    Negative terminal vs Ground. Double definitions?

    In a regular household plug-in, there are 3 prongs: positive, negative, and ground. But, when doing wiring projects, many times they refer to the negative lead as the ground. Why is it that this lead is sometimes referred to as the ground, while other times, the ground is a completely separate...
  38. nomadreid

    Conflicting definitions of temperature?

    I thought that temperature is a measure of energy density, which means that at the vacuum energy has a minuscule temperature above absolute zero. However, I read at http://www.Newton.dep.anl.gov/Newton/askasci/1993/physics/PHY59.HTM that "At absolute zero, all motion does not cease,..." which...
  39. D

    Comparing Tensor Double Dot Scalar Product Definitions

    Ok I have seen the tensor double dot scalar product defined two ways and it all boils down to how the multiplication is defined. Does anyone know which is correct? I believe the first one is correct but I keep seeing the second one in various books on finite element methods. 1. \nabla \vec{u}...
  40. S

    Sidereal Time: Clarifying Contradictory Definitions

    Help! I need some clarification on definitions, because it seems like I am getting contradictory information. My textbook defines sidereal time as simply the right ascension that is on the local meridian. It further defines sidereal time as being the RA of a star + the hour angle of the...
  41. W

    Chemistry Crossword Definitions

    1. 7.Across: A piece of iron equipment attached to a retort stand to support a gauze or hold a funnel. Down: A metal rod to which you attach clamps and other equipment 2. none 3. This is part of a crossword and am not given any words can you help me find these words because my...
  42. W

    Is the chemical used with caution in home and lab, could it be acid?

    1. 1. a.Metal mesh used to prevent glassware from cracking when heated with a Bunsen burner b. A type of chemical which must be respected and handled carefully in the home and laboratory. 2.None 3. I have to figure out what the word is because its a crossword puzzle and it does...
  43. Rasalhague

    Definitions of operation and function

    Definitions of "operation" and "function" Is every operation a function? Is every function an operation? From the definitions I've read, I'm guessing yes. If not, what would be an example of a function that isn't an operation, or an operation that isn't a function?
  44. DrGreg

    Published definitions of rapidity

    Rapidity of a particle with speed v can be defined as either c tanh-1(v/c).....(1) or tanh-1(v/c).....(2) The difference is that (2) is a dimensionless hyperbolic angle whereas (1) has the same dimensions as speed (and is nearly equal to speed when small). Equivalent definitions...
  45. L

    What are the two definitions of gamma function and how are they related?

    I spend some time studying special functions recently. I found two definitions of gamma function, one in form of integral and the other in form of infinite products, and I cannot prove of their equivalence. I found the definition in infinite product form important in proofing many properties of...
  46. D

    Why Are Electric and Magnetic Fields Defined Differently in SI Units?

    Why is the electric field defind as per meter while the magnetic field is defined as per square meter? Does Ampere's law only contain the current density in the solution because of this fact? Couldn't your simplify the Maxwell equations if you changed the SI definitions?
  47. Fredrik

    Equivalent definitions of continuity (topological spaces)

    Not really homework, but a typical exercise question, so I figured it's appropriate to post it here. Homework Statement X,Y topological spaces f:X→Y x is a point in X Prove that the following two statements are equivalent: (i) f^{-1}(E) is open for every open E that contains f(x)...
  48. L

    Are all iff s definitions?

    are all "iff"s definitions? are all statements of the form "p if and only if q" definitions or equivalences? can there be any iff statements that are not statements of equivalence?
  49. S

    Speaking of concepts where CLEAR definitions

    Was in the shower and I was thinking of this conversation I was having. They were completely ignorant to things as they really are. Instead they took them as how they THINK they are. i.e. science. I assume he does this so that the concepts can fit to whatever he is arguing for or so that he...
  50. Rasalhague

    Definitions of the Lagrangian and the Hamiltonian

    I've just encountered the terms Hamiltonian and Lagrangian. I've read that the Hamiltonian is the total energy H = T + U, while the Lagrangian L = T - U, where T is kinetic energy, and U potential energy. In the case of Newtonian gravitational potential energy, U = -G\frac{Mm}{r}. So am I...
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