Results 1 to 8 of 8

Math Help - check my proof f(phi) = phi

  1. #1
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    198

    check my proof f(phi) = phi

    Hi

    I need to check my proof. Suppose that \mathbi{f}:\mathbi{X}\rightarrow\mathbi{Y}

    I have to prove that \mathbi{f}(\phi)=\phi

    now here's my proof. Let \small y_1\in f(\phi) be arbitraty
    \exists\ x_1\in \phi \ni f(x_1)=y_1
    But since \phi is an empty set, there is no element x_1 in it. So function operation f(x_1) can't be performed. In other words, y_1\in \phi . This is one way proof.
    Is the proof ok so far ? I also need to prove in other direction, but I am confused about it.

    thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2010
    Posts
    466
    Thanks
    4
    I take it that you're using the letter phi to stand for the empty set.

    As far as I can tell, you must have left out something in the statement of the problem.

    The following is NOT a theorem (of, say, set theory), so I don't know why you would be asked to prove it [edit: '0' for the empty set]:

    If f:X->Y, then f(0) = 0.

    Counterexample:

    Let X = {0}, let Y = {1}, let f = {<0 1>}.

    Then, also, your "proof" makes no sense. There's no basis to assume there is an x1 such that x1 is in 0 and/or f(x1) in 0.

    What book is this problem taken from?
    Last edited by MoeBlee; October 15th 2010 at 08:52 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,417
    Thanks
    718
    The LaTeX symbol for the empty set is \emptyset.

    What do you mean by f(\emptyset)? Is it the case that \emptyset\in X, so f is applied to an element of its domain? Or is it a special case of f(A) where A is a subset, not an element of X? This latter thing, also sometimes denoted by f[A], is defined as \{f(x)\mid x\in A\} and is a subset of Y. In this case, f(\emptyset)=\emptyset by definition, because there is no x in \emptyset that you can apply f to.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    198
    Hi

    I have taken the problem from "A friendly introduction to analysis:single and multivariable " second edition by Witold Kosmala. And the statement is

    Suppose that f:X\rightarrow Y Prove that
    f(\emptyset)=\emptyset where \emptyset is an empty set.

    Since both f(\emptyset)=\emptyset and \emptyset are sets, I am using
    standard approach used to prove the equality of the sets. I have downloaded the
    errata from the author's website at
    Code:
    http://www.mathsci.appstate.edu/~wak/
    thanks
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    198
    Hi emakarov

    I have stated the complete problem. that's all the author has given. this is the first chapter in the book and in this section, he is covering functions. even I am not sure what he is trying to say

    newton
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Feb 2010
    Posts
    466
    Thanks
    4
    And that symbol is ambiguous in general mathematical writing unless the author specifies it means 'subset' or 'proper subset', since some authors use it to mean 'subset' and other authors use it to mean 'proper subset'.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Feb 2010
    Posts
    466
    Thanks
    4
    There must be some context missing here.

    I gave a counterexample already.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    198
    Quote Originally Posted by MoeBlee View Post
    There must be some context missing here.

    I gave a counterexample already.
    thats what I think. some information missing. thanks by the way
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof check please: Two DFA's
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 28th 2011, 09:37 PM
  2. Proof check
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: July 14th 2010, 06:25 AM
  3. Please check my proof
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 20th 2009, 12:19 PM
  4. Check my proof
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 9th 2008, 10:34 PM
  5. Check this proof.
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 4th 2006, 02:56 PM

/mathhelpforum @mathhelpforum