1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

'Is $ℜ$ an equivalence relation?

  1. Apr 28, 2013 #1
    1. The problem statement, all variables and given/known data

    The **mean value theorem** says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u).$$ Here is my question.

    Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u);$$ assume that $f$ is a non-zero analytic function in the whole real line.

    We can define the relation $ℜ$ by
    $$cℜd⇔f′(c)=\frac{1-t}{1-u}f′(d)$$

    where $u∈ℝ$ such that $f(u)=0$, $c∈(u,v)$ and $t∈ℝ$ such that $f(t)=0$, $d∈(t,v)$, that is we apply the mean value theorem for f in the two intervals $(u,v)$ and $(t,v)$, here $u,t,v$ are arbitrary.

    Is $ℜ$ an equivalence relation? If so, determine its equivalence classes.

    2. Relevant equations



    3. The attempt at a solution
    I have no idea to start with.
     
  2. jcsd
  3. Apr 28, 2013 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I am confused by your "Assume that u is a root of f", "assume that f is a non-zero function in the whole real line" and "where u such that f(u)= 0".
     
  4. Apr 28, 2013 #3

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    "non-zero function" is equivalent to "NOT (f(x)=0 everywhere)". And the function is defined in the whole real line.

    I am confused by that t, however. Do we have some constant t? Are we free to choose t for each pair (c,d) to analyze? What about d, can we choose that as well?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: 'Is $ℜ$ an equivalence relation?
  1. Equivalence relation (Replies: 4)

  2. Equivalence relations (Replies: 1)

  3. Equivalence Relations (Replies: 11)

  4. Equivalence Relations (Replies: 17)

Loading...