What is Definition: Definition and 1000 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. N

    I By Continuity definition 1/0 is infinity

    lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continuous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?
  2. Mr Davis 97

    I Definition of the root of -1 for different roots

    How does the value of ##\displaystyle \sqrt[a]{-1}## vary as ##a## varies as any real number? When is this value complex and when is it real? For example, we know that when a = 2 it is complex, but when a = 3 it is real. What about when ##a = \pi##, for example?
  3. Mind----Blown

    Surface current density-logic and definition contradict

    Okay, so in Griffith's introduction to electrodynamics, Griffith clearly defines surface current density as follows: "when charge flows over a surface, we describe it by the surface current density, K. Consider a 'ribbon' of infinitesimal width dL running parallel to the current flow. If the...
  4. Mr Davis 97

    I Definition of absolute value

    If we define ##|a| = \sqrt{a^2}##, then why can't we do something like ##\sqrt{a^2} = (\sqrt{a})^2 = a##? Or equivalently ##\sqrt{a^2} = (a^2)^{1/2} = a^{2/2} = a##? Isn't this a contradiction? Also, how would this relate to showing that ##\sqrt{|a|} = |\sqrt{a}|## is true or false?
  5. S

    I Understanding the Definition of Boundary in Set Theory for Topological Spaces

    Hello! This is more of a set theory question I guess, but I have that the definition of the boundary of a subset A of a topological space X is ##\partial A = \bar A \cap \bar B##, with ##B = X - A## (I didn't manage to put the bar over X-A, this is why I used B). I think I have a wrong...
  6. H

    Understanding Moment and Center of Gravity in One Dimension

    Homework Statement The book I am reading says that given a point mass(m) at the point x, the quantity mx is the "moment about the origin)" It then defines the moment of a collection of points as M = m(1)x(1) + m(2)x(2) + ... m(n)x(n) where m(1) = mass of first point and x(1)=distance of first...
  7. S

    I Definition of Topology - What Does {##U_\alpha | \alpha \in I##} Mean?

    Hello! I just started reading an introductory book about topology and I got a bit confused from the definition. One of the condition for a topological space is that if ##\tau## is a collection of subsets of X, we have {##U_\alpha | \alpha \in I##} implies ##\cup_{\alpha \in I} U_\alpha \in \tau...
  8. R

    I Measuring Vectors: Defining Direction and Magnitude

    On wiki measurement is defined as: Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. So a number. For scalars, I agree because they are described by a single number. But what about vectors? They have...
  9. F

    Is this definition of Reynolds' Transport Theorem correct?

    One thing I find frustrating when trying to get a handle on this theorem is the number of different forms presented in the literature. I understand this to be due to it being very general theorem applicable to many different contexts. Not that the world needs a new, slightly different looking...
  10. W

    Stargazing [just for fun] Planet Definition

    So I'm reading in the news and a group of scientists from the New Horizons missions appear to be reopening "The Pluto Debate" with yet another new definition of planet at the Lunar and Planetary Science Conference in March:rolleyes:. If accepted their definition of planet would increase the...
  11. Mr Davis 97

    I Generalizing the definition of a subgroup

    Let ##G## be a group. I have shown that ##H_a = \{x \in G | xa=ax \}## is a subgroup of G, where ##a## is one fixed element of ##G##. I am now asked to show that ##H_S = \{x \in G ~| ~xs=sx,~ \forall s \in S\}## is a subgroup of ##G##. How would proving the former differ from proving the latter...
  12. T

    How do I define haploid and monoploid?

    I think I understand these terms, not fully though and I can't distinguish between the two and define them.
  13. Ravik Rocha

    MHB General definition of parameter

    I'm reading a very general definition of parameter on this site Parameter definition - Math Insight. I didn't understand why we call the variable $t$ of the curve $\alpha(t)=(\cos t,\sin t)$ a parameter. For me $t$ in this case is a variable too according to the definition of the site I...
  14. fisher garry

    Definition of stress and usage of normal vector

    Homework Statement The texts are taken from http://ingforum.haninge.kth.se/armin/fluid/exer/deriv_navier_stokes.pdf and https://simple.wikipedia.org/wiki/Stress_(mechanics)Homework EquationsThe Attempt at a Solution The formula for stress is ##\sigma=\frac{F}{A}## (I). From the document...
  15. Math Amateur

    MHB Understanding Irreducible Elements in Integral Domains - Peter's Questions

    Joseph A. Gallian, in his book, "Contemporary Abstract Algebra" (Fifth Edition) defines an irreducible element in a domain as follows ... (he also defines associates and primes but I'm focused on irreducibles) ... I am trying to get a good sense of this definition ... My questions are as...
  16. Math Amateur

    I Definition of an irreducible element in an integral domain

    Joseph A. Gallian, in his book, "Contemporary Abstract Algebra" (Fifth Edition) defines an irreducible element in a domain as follows ... (he also defines associates and primes but I'm focused on irreducibles) ... I am trying to get a good sense of this definition ... My questions are as...
  17. Andy_K

    I Is Time Merely the Progression of Energy Toward Equilibrium?

    Can we say that TIME is essentially the progression of energy from one state to another, in its long cosmic quest to achieve equilibrium? Without the movement of electromagnetic waves and transfer of energy, the entire universe would come to a standstill — an inanimate, “frozen” world. Time...
  18. Math Amateur

    MHB Thinking about the Definition of a Unit of a ring R .... ....

    I have been thinking around the definition of a unit in a ring and trying to fully understand why the definition is the way it is ... ... Marlow Anderson and Todd Feil, in their book "A First Course in Abstract Algebra: Rings, Groups and Fields (Second Edition), introduce units in a ring with 1...
  19. Math Amateur

    I Thinking about the Definition of a Unit of a ring R .... ....

    I have been thinking around the definition of a unit in a ring and trying to fully understand why the definition is the way it is ... ... Marlow Anderson and Todd Feil, in their book "A First Course in Abstract Algebra: Rings, Groups and Fields (Second Edition), introduce units in a ring with 1...
  20. A

    I The space-time interval definition in special relativity

    Hello, please note that the following is only about special relativity, not general. Of course, if there are any things to point out that fall in general relativity, feel free to do so, but I don't know GR, so I won't understand arguments based in GR. I also am not great with a geometry-based...
  21. evinda

    MHB Meaning of Definition: $T_n(1)$ Defines a Function

    Hello! (Wave) I want to show that the language $L=\{ n \in \mathbb{N}_0 \mid T_n(1) \text{ defines a totally defined function }\}$ is not decidable. What does it mean that $T_n(1)$ defines a totally defined function? Does it mean that $T_n(x)$ is a totally defined function and $1$ belongs to...
  22. binbagsss

    I Definition of discrete Subgroup quick q

    Hello, Just a really quick question on definition of discrete subgroup. This is for an elliptic functions course, I have not done any courses on topology nor is it needed, and most of the stuff I can see online refer to topology alot, so I thought I'd ask here. I need it in the complex plane...
  23. S

    I Epsilon in the limit definition

    in the limit definition of a sequence, why does epsilon has to be greater than 0 and not greater or equal to 0? thanks in advance.
  24. P

    The right definition of magnetic field?

    I thought the definition was: a vectorial field that interacts with a moving charge But wikipedia says it's a field generated by electric currents and magnetic materials... https://en.m.wikipedia.org/wiki/Magnetic_field Why is in the definition the way it is Generated? When we had defined the...
  25. H

    MHB Rigorous definition of "Differential"

    First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else. Now let's get started: When it comes to definitions, I will be very strict. Most textbooks...
  26. G

    Rigorous definition of magnetic poles?

    Hi. Since there haven't been observed magnetic monopoles so far, what exactly do we mean when we talk about the north/south pole of a magnet? Is it something like "north is where the field lines exit a solid body" and "south is where they enter" or is there a more rigorous definition?
  27. C

    I Towards a matrix element definition of PDF

    In Schwartz's book, 'Quantum Field Theory and the Standard Model' P.696, he starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. Most of the derivation is clear to me, except a couple of things at the start and midway. The...
  28. Z

    Momentum not consistent with definition in Landau's book?

    Homework Statement I am not sure whether the meaning of the equation ##(3)## which used for deriving momentum is as same as equation ##(4)##.I will make a detailed description below. The lagrangian function for a free particle is ##L=-mc^2\sqrt{1-\frac{v^2}{c^2}} \quad (1)## The action from...
  29. N

    MHB Chemistry: Demonstrate the definition with a chemical equation for HCl.

    Demonstrate the definition with a chemical equation for HCl. Express your answer as a chemical equation. Identify all of the phases in your answer. My Answer: HCl(g)+H2O(l)→H3O+(l)+Cl−(aq)What the answer said: There is an error in your submission. Make sure you have formatted it properly.
  30. S

    A What are the different decay processes shown in this Feynman diagram?

    Consider the following momentum-space Feynman diagram https://upload.wikimedia.org/wikipedia/commons/b/b5/Lepton-interaction-vertex-eeg.svg This Feynman diagram is the leading-order contribution to the any of the following processes: 1. ##e^{-} \rightarrow e^{+} + \gamma## 2. ##e^{+}...
  31. S

    I Spivak's Definition of Integrals vs Stewart's

    Hello. I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my...
  32. V

    Consider the definition of 1m

    Consider the definition of 1m. It is the distance traveled by light in 1 / 299,792,458 seconds (lets call this X). Fair enough a definition. Now let's assume, for an instant, that somewhere in our past, while scales were being developed, man erroneously decided that instead of the distance...
  33. W

    Statistical definition of temperature

    hello everyone. i need help understanding this statement: d(lnΩ)/dE = 1/kbT so Ω are the posible microstates for energy E, and the derivative of Ω w.r.t E is 1/kbT. why? what i understand so far is: looking at the division of energy of two "connected" systems the energy will divide itself in a...
  34. Euler2718

    Linear Dependence and Non-Zero Coefficients

    Homework Statement True or False: If u, v, and w are linearly dependent, then au+bv+cw=0 implies at least one of the coefficients a, b, c is not zero Homework Equations Definition of Linear Dependence: Vectors are linearly dependent if they are not linearly independent; that is there is an...
  35. Isomorphism

    On the definition of width in Doppler broadening problems.

    The following question has stumped me. I am not getting the answer marked. Instead I am getting option a). I have produced my attempt below. Also I have no idea which formula to use when the question says Doppler width. Since frequencies form a distribution, there are multiple notions of width...
  36. L

    I What is the standard unit of mass and why is it defined in terms of an artifact?

    The standard unit of mass is defined to be "equal to the mass" of a cyliner of platinum and iridium in france. I have always wondered what this means? Why are all sources saying mass is defined in terms of mass? What were the steps taken to decide this unit? I know that base quantities have to...
  37. Z

    I Help please -- Seeking clear definition of fundamental terms

    I've tried repeatedly to get clear, succinct definitions of the following terms over and over again, but invariably the definitions provided clash, and I'd like to put an end to that. The terms I am trying to define clearly are: - Relation - Definition (Mathematical definition) - Function -...
  38. S

    I Is the definition of a phase arbitrary?

    Gibbs phase rule says f = r-M+2 with f: thermodynamic degrees of freedom; r: number of components; M: number of phases I wonder whether the defintion of "phase" is restricted or almost arbitrary. For example, consider a system of H2O, O2 and H2 in a closed vessel. Let there be the contstraint...
  39. F

    Maple Maple - Explicit Set Definition & Natural Number Symbol

    Hi, I would like to know if is possible define a set rule explicitly. ex.: Define the set A explicitly where ξ = ℕ and A = {x | x < 3}. The maplesoft said the (N) is a representation of Natural Numbers. f:=proc() local a,b; a:=N; b:= 3; print(a<b); end proc; when I...
  40. L

    B Clear definition of microgravity?

    Hi, I read about definition of microgravity. It is usually described as reduced g, but not zero g. How can one say then that an object is in microgravity? I was looking hours for a clear definition, like an object is in microgravity if there are just 10^-6g left (clearly wrong, because I read...
  41. S

    A Definition of the Lie derivative

    Consider the Lie derivative of the vector field ##\bf{Y}## with respect to the vector field ##\bf{X}## on manifold ##M^{n}(x)## defined as ##\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}## Now, I understand that...
  42. F

    I Definition and measurement of proper length

    As I understand it, the proper length, ##L## of an object is equal to the length of the space-like interval between the two space-time points labelling its endpoints, i.e. (in terms of the corresponding differentials) $$dL=\sqrt{ds^{2}}$$ (using the "mostly plus" signature). Furthermore, this is...
  43. Mayan Fung

    Can we define a circular loop with moving electrons as magnetostatic?

    When I learned magnetostatics. My teacher and book said that it is the case of steady current. However, if I consider a circular loop, the electrons are in fact moving in uniform circular motion. That means they are accelerating. How come we can still define it to be a magnetostatic situation
  44. J

    I Help to understand second order differential definition

    Hi folks, I am reading Poisson's Teatrise on Mechanics. In the introduction he talks about the infinitesimals. Let's say A is a first order infinitely small quantity, a differential of the first order, if the ratio of A to B is infinitely small too it means B is an infinitesimal of the second...
  45. Metals

    B A statistical definition of Young's Modulus?

    Young's Modulus is usually defined as the intrinsic property of a material indicating it's stiffness, or it's ability to resist deformation. Though, it is measured in Pa, meaning it should have some statistical description. Spring constant, for example, can be define as the stiffness of an item...
  46. S

    What is the Difference Between Gravitational Force and Weight?

    Homework Statement Centre of gravity - the point at which: 1) gravity acts on a body or 2) weight of a body may be considered to act. The answer is 2) and I understand why - because gravity acts all over but it is easier to calculate a single point, an average point of where the mass is...
  47. M

    Theoretic doubt about the definition of derivatives.

    Homework Statement Hi, this is a question that has been bothering me for a while. (Im in calculus II at the moment) Why do i need to derivate some functions by definition and other times i dont? for example if somebody asks me to calculate the partial derivatives of a branch function in a a...
  48. dbc

    Let's change the definition of 'weight'

    Since 97% of everyday weight scales (both in doctor offices and at home) measure our actual mass either in lbm and/or kg, and NOT force (lbf or N), then why does oxford choose to define weight to be relative? "a body's relative mass or the quantity of matter contained by it, giving rise to a...
  49. Minestra

    Definition of Success problems with projectile motion

    So I just got beat up by this question on my midterm. I'm not sure if these problems are always called definition of success but that is how my professor refers to them as. The question: (paraphrased) When you walk into your dorm room you like to throw your keys onto the center of your desk...
  50. A

    Exploring the Mathematical Definition of Time

    Is there a good mathematical definition of time?
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