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.

View More On Wikipedia.org
Recent contents Top users
  1. L

    Intuition about definition of laplace transform

    why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...
  2. genphis

    Is the definition of space relative ?

    I have just finished reading stuart clark's book 'The Universe' and i find myself pondering the question of space its possible infinite size,shape, and its relation to our universe. a) if space did not exist before our universe's expansion. What are we expanding into and what are we pushing...
  3. S

    What does the N mean in a Cauchy sequence definition?

    What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy...
  4. C

    Relativistic at freeze out? Definition of HDM

    Okay so in a HDM scenario, I have seen it described that the neutrinos were relativistic at freeze out. (If I could find it I would reference it.) Is this a contradictory statement? The condition for relativistic travel is E>>m but just before freezeout, the neutrino has energy equal to...
  5. C

    Vector Definition: Explaining Left Side to Right Side

    Hi For some strange reason I just can't see why this is true?? Can anyone help me explain why the left side can be written as the right side? I added a picture.
  6. B

    Exploring the Definition of Space in Physics

    There is any space ? In physics space has any definition ?
  7. S

    Definition of a Fractional Derivative/ Integral

    The geometric and physical properties of derivatives and integrals to an integer order are easy to describe, but fractional calculus is obviously present in modern mathematics and physics. That being said, are there a generalizations of the definitions derivatives and integrals that include...
  8. B

    Dirac's equation notation definition?

    Homework Statement I've been asked to write what this is explicitly - γμ∂μ Homework Equations (γμ∂μ-im)ψ = 0 - Dirac's equation The Attempt at a Solution I understand that γμ is a matrix but depending on what μ is they're all completely different :S How can it be written...
  9. O

    Definition of first law of thermodynamics

    The total work is the same in all adiabatic processes between any two equilibrium states having the same kinetic and potential energy. That is another way to describe first law of thermodynamics , and define internal energy. My question is what does "the same kinetic and potential energy "...
  10. P

    How Is Flow Velocity Defined in Fluid Dynamics?

    Good day, In my book, the following definition for flow velocity is given: So summarized, the flow velocity at a point in space is the velocity of an infinitesimally small fluid element as it sweeps through that point. But now my question; how is the velocity of an infinitesimally small...
  11. G

    Definition of entropy of complex systems is existed?

    Common extensive quatities such as mass, charge, volume can be defined for general systems. I can imagine that we can measure and define them without any problem in case of any kind of complex system as well. However, I do not know the general definition of the entropy, only the thermodynamic...
  12. skate_nerd

    MHB Sequence with recursive definition?

    Sorry to spam my problems all over this forum but series have me struggling somewhat. Last problem on my homework is the sequence an defined recursively by: a1=1 and an+1= \(\frac{a_n}{2}\) + \(\frac{1}{a_n}\) First part was the only part i know how to do. it was to find an for n=1 through 5...
  13. P

    Use L'Hopital's Rule to relate to limit definition for e

    Homework Statement It can be shown that lim n→∞(1 + 1/n)^n = e. Use this limit to evaluate the limit below. lim x→0+ (1 + x)^(1/x) Homework Equations The Attempt at a Solution So i guess what i need to do is try to get that limit in the form of the limit definition for e...
  14. M

    Can Variable Message Lengths Invalidate Cryptographic Indistinguishability?

    Prove that the following definition cannot be satisfied if Π can encrypt arbitrary-length messages and the adversary is not restricted to outputting equal-length messages in experiment PrivKeavA,∏. A prive-key encryption scheme ∏=(Gen, Enc, Dec) has indistinguishable encryptions in the...
  15. D

    Are All Inflection Points Also Critical Points?

    Are inflection points critical points? and what about at the value that f(x) undefined? Is that critical point too?
  16. M

    A simple definition to Non-Relativistic Quantum mechanics?

    What is it in a simple definition, and how does it differ from relativistic quantum mechanics?
  17. F

    Prove limits using epsilon delta definition

    Homework Statement http://store2.up-00.com/Sep12/JB498124.jpg 2. The attempt at a solution No attempts because i can't understand how to solve it
  18. S

    What is the Definition of the Supremum in First Order Predicate Logic?

    i was trying to formalize the definition of the supremum in the real Nos (supremum is the least upper bound that a non empty set of the real Nos bounded from above has ) but the least upper part got me stuck. Can anybody help?
  19. V

    Confused about definition of wavelength

    Wavelength of a sinusoidal wave is defined as the spatial period of the wave, which can be measured between any two points on the wave where the shape repeats. But if my wave is defined as a function of time (like those of a simple harmonic oscillator), how can you say the distance along the...
  20. S

    Is the proof that ##0^2>0## false if a =0 possible?

    Which of the following two definitions is correct: 1) ##\forall x\forall y[ |x|=y\Longleftrightarrow( x\geq 0\Longrightarrow x=y)\wedge(x<0\Longrightarrow x=-y)]## 2) ##\forall x\forall y[ |x|=y\Longleftrightarrow( x\geq 0\Longrightarrow x=y)\vee(x<0\Longrightarrow x=-y)]## I think the...
  21. R

    Questions Regarding Definition of One-to-One and Onto Functions?

    Hi, I was just having a little trouble of understanding what it... is saying, well first I'll state what my book says the definition is: A function T:D* \subseteq R2 → R2 is called one-to-one if for each (u,v) and (u',v') in D*, T(u,v)=T(u',v') implies that u = u' and v = v' A function...
  22. binbagsss

    Function Definition / Concept - Codomain, range, domain etc.

    What must actually be specified in order for a function to be fully defined / or in what combinations if not all 3 need to be specified? I.e - from knowing the function you can determine the co-domain - e.g - if it is specified that real functions are going in, and for something simple like 2x...
  23. E

    Understanding Modules: Definition and Properties for Homework

    Homework Statement I am curious if all modules contain 0. Homework Equations A left R-module M over the ring R consists of an abelian group (M, +) and an operation R × M → M such that certain properties hold... The Attempt at a Solution The definition of a module says that it is an...
  24. R

    Definition of the dielectric function in the linear response regime

    Sometimes the dielectric function is defined as the connection between the total electric field in a material and the external field, \mathbf{E}(\mathbf{r},\omega) = \int \epsilon^{-1}(\mathbf{r},\mathbf{r'},\omega) \mathbf{E}_{\text{ext}}(\mathbf{r'},\omega) d \mathbf{r'}, and sometimes...
  25. M

    Epsilon Delta Limit Definition

    Homework Statement Prove lim x--> -1 1/(sqrt((x^2)+1) using epsilon, delta definition of a limit Homework Equations The Attempt at a Solution I know that the limit =(sqrt(2))/2 And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if...
  26. A

    Redundancy in definition of vector space?

    According to my book, a vector space V is a set endowed with two properties: -closure under addition -closure under scalar multiplication and these two properties satisfy eight axioms, one of which is: "for all f in V there exists -f in V such that f+(-f)=0" But then isn't this axiom...
  27. A

    Formal definition of limits as x approaches infinity used to prove a limit

    Homework Statement use the formal definition to show that lim as t goes to infinity of (1-2t-3t^2)/(3+4t+5t^2) = -3/5 Homework Equations given epsilon > 0, we want to find N such that if x>N then absolute value of ((1-2t-3t^2)/(3+4t+5t^2) + 3/5) < epsilon The Attempt at a Solution...
  28. STEMucator

    Multivariable limit definition question

    Homework Statement I'm reading through Taylor's advanced calculus and came across this question in section 7.2 : http://gyazo.com/6b0c5a2e4e605ff77bf6584eb3295948 Homework Equations The definition of the partial of f with respect to some variable at some point (a,b), let's say the...
  29. R

    Rigorous dirrerential definition and physics

    My question is rather simple but it puzzles me for a long time actually. If we have a look at differential as physicists usually do we came up with a simple definition of "infinitesimal variable change". And this idea then preserves elsewhere like in the definition of entropy: \mathrm{d} S =...
  30. Z

    How to prove the definition of arctangent function using integral?

    Homework Statement This is a problem from Introduction to Analysis by Arthur P. Mattuck,chapter 20,problem 20-1. <a href="http://www.flickr.com/photos/86024731@N04/8090259684/" title="arctangent by gnu is not unix, on Flickr"><img...
  31. D

    Precise definition of limits at infinity

    Homework Statement Let f be a continuous function on ℝ. Suppose that \mathop {\lim }\limits_{x \to - \infty } f(x) = 0 and \mathop {\lim }\limits_{x \to \infty } f(x) = 0. Prove that there exists a number M > 0 such that \left| {f(x)} \right| \le M for all x \in ℝ. Homework Equations...
  32. A

    Finding the Derivative of f(a) with Definition

    Homework Statement Find derivative of f(a) for f(t)=(2t+1)/(t+3) using the definition of a derivative Homework Equations f '(a)=lim as x goes to a of (f(x)-f(a))/(x-a) The Attempt at a Solution f '(a)=lim as x goes to a of (f(x)-f(a))/(x-a) f '(a)=lim as t goes to a of...
  33. M

    What does this definition mean ?

    hello , I don't understand the meaning of the next definition , so , I hope that you can make it easy to understand it for me definition if G is an arbitrary group and ∅≠S⊆G , then ,the symbol (s) will represent the set (S) = ∩ { H∖S ⊆ H : H is a subgroup of G } can you give me some...
  34. C

    Proof of lim(1/x) x->0 by negating epsilon delta definition of limit

    Homework Statement I want to show that \lim_{x \rightarrow 0}\frac{1}{x} does not exist by negating epsilon-delta definition of limit. Homework Equations The Attempt at a Solution We say limit exists when: \forall \epsilon > 0, \exists \delta > 0 : \forall x(0< \left| x\right| < \delta...
  35. N

    Definition of electrical branch

    Hi everyone! I wanted to know why in the circuit analysis a generator doesn't represent an electrical branch? And the second question is if two resistors are in series on a wire, does it represents only a branch(the series of the resistance) or two branches ?
  36. M

    Why we use strictly less than delta and epsilon in definition of limits

    Homework Statement I'm wondering why we can't use less than or equal to for the formal definition of the limit of a function: Homework Equations lim x→y f(x)=L iff For all ε>0 exists δ>0 such that abs(x-y)<δ implies abs(f(x) - L)<ε Why not: lim x→y f(x)=L iff For all ε>0 exists...
  37. B

    Definition Of The Complement Of A Set

    Hello, my book defines the complement of a set like so: {x ∈ U | x /∈ A} To me, it seems like the definition should be (x| x \in U \wedge x \notin A) Which is more proper?
  38. K

    What is the Definition of Closed Sets in Topology?

    Good day! Im currently reading the book of Steven R. Lay's "Analysis with an Introduction to Proof, 3rd ed.". According to his book, if a subset S of ℝ contains all of its boundary then it is closed. But i find this wrong since if we consider S={xεQ;0≤x≤2}, then it can be shown that S...
  39. S

    The definition of the density operator in Pathria

    Hello Everybody, I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator: \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, where N is the number of systems in the ensemble and the...
  40. S

    Using Limit Definition of the Derivative?

    If one uses the limit definition of a derivative (lim of (f(x)-f(a)) / (x-a)) as x approaches a) on a function and you get a value (ie. it is not undefined) does that mean the derivative of the function at that point exists? In other words, even if the limit definition of the derivative works...
  41. D

    Understanding Equality in Mathematics: A Comprehensive Guide

    I'm particularly interested in the foundation of mathematics. I've read various ways of defining the integers and addition. But I never encountered a formal definition of equality. It's seems (at least for what i read) that the equality is treated as something fundamental that does not need to...
  42. S

    What is the definition of entropy in SM?

    We physicists must be careful to insure that theories begin with correct principles. One basic principle is that all quantities must be capable of being observed or measured. If a theory uses a quantity that cannot be observed, then it is not a physics theory, but a hypothesis or a...
  43. C

    Definition of additive inverse and operation of negation.

    Hello everyone. I am slightly confused by these ideas so i would like your help. How is additive inverse defined? Is unary negation an operation in its own right just like those more familiar, like addition, multiplication? Or something else?
  44. A

    Definition of relative error

    Hi all, I have a general question about relative error. Suppose that we have a vector of measurements \hat{b}=\left(\hat{b_{1}},\hat{b_{2}},...,\hat{b_{n}}\right). Furthermore, suppose that these measurements are accurate to 10%. My natural interpretation of this statement is that there is...
  45. G

    Exploring the Pauli Vector: Mathematical Definition

    Hello I'm reading my old notes of QM, I found the definition of Pauli vector, as follow \vec{\sigma}=\sigma_1 e_x+\sigma_2e_y + \sigma_3 e_z Where e_x. e_y and e_z are unit vectors. So, here is my question. \sigma_i and e_i are elements of different nature. How can we define the product...
  46. D

    Proving limit statement using delta-epsilon definition

    Homework Statement For all x\in R , f(x)>0 . Using precise definition of limits and infinite limits, prove that \lim_{x\to a}f(x)=\infty if and only if \lim_{x\to a}\frac{1}{f(x)}=0 Homework Equations The Attempt at a Solution I know the precise definition of limits and infinite limits...
  47. P

    Rigorous definition of a limit

    First off I want to apologize for bombarding this subforum with my gazillion questions. If my continuous barrage of questions poses a problem just let me know and I'll stop. Homework Statement For each value of ε, find a positive value of δ such that the graph of the function leaves the window...
  48. T

    Definition of gene needs reveiwing ?

    According to this news article , genome analysis has shown that the understanding of what constitutes a gene has to reveiwed and redefined in light of new evidence. Btw there is no junk DNA in our genome. http://medicalxpress.com/news/2012-09-encode-massive-genome-analysis-gene.html
  49. P

    Leibniz's Rule Proof With Definition of a Derivative

    Homework Statement Use the definition of the derivative to show that if G(x)=\int^{u(x)}_{a}f(z)dz, then \frac{dG}{dx}=f(u(x))\frac{du}{dx}. This is called Leibniz's rule. Also, by thinking of the value of an integral as the area under the curve of the integrand (and drawing a picture of...
  50. Z

    Definition of function domain and range

    I am a bit confused about this matter. While I was studying Calculus I saw an excercise like this: The domain of f(x) [0,2] and the range is [0,1], it also shows its graphic, though it is not important it is something like a parabola, its maximum point is (1,1) and its intersection points are...
Back
Top