Results 1 to 9 of 9

Math Help - Need help proving Theorem dealing with functions

  1. #1
    Newbie
    Joined
    Nov 2010
    Posts
    4

    Need help proving Theorem dealing with functions

    Hey everyone - I'm having trouble proving a theorem that I don't think should be too hard to prove:

    Theorem (Equality of Functions):
    If f: A -> B and g: A -> B are functions, then f=g if and only if f(a) = g(a) for all a that are an element of A.

    Any help would be greatly appreciated.

    Thanks,
    Robert
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    What is your definition for the equality of functions?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2010
    Posts
    4
    I believe that "equality of functions" just refers to what is being said in the theorem - that two functions are equal iff f(a) = g(a)

    The thing that I really need to do is prove the statement.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    I believe that "equality of functions" just refers to what is being said in the theorem - that two functions are equal iff f(a) = g(a).
    That sentence makes no sense. A theorem is something you prove. If you are to prove this theorem, then it must be that you have a different definition of function equality that you're trying to show is equivalent to the one in the theorem. Otherwise, you have a definition here, not a theorem; you don't prove definitions.

    I'm used to thinking of the equality of functions this way: two functions are equal if and only if their domains are the same, and their rules of association are the same. And this is precisely what your theorem is saying. Therefore, in order to help you out, I need to know how exactly you are defining function equality.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    It might also be appropriate if you tell us exactly how you defined "function" in our class. that just might be all we need. but yes, if you have any alternative definition of equality of functions, you should state that as well. It shouldn't be difficult once we know what definitions we can use.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Nov 2010
    Posts
    4
    Sorry about that - this is how it is written in the book:

    "Two functions are equal if they have the same domain, the same codomain, and "agree" on every element of the domain. More formally,

    Theorem: If f: A -> B and g: A -> B are functions, then f=g if and only if f(a) = g(a) for all a that are an element of A.

    (Hint: a function is a relation. Saying that two functions are equal is to say they are relations between the same pair of sets, and they include the same ordered pairs.)"
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Nov 2010
    Posts
    4
    One more definition to work with:

    Definition: let A and B be nonempty sets. A function f from set A to set B (denoted by f: A -> B) is a relation between A and B satisfying the following conditions:
    1. For each a that is an element of A, there exists b that is an element of B such that (a,b) is an element of f, and
    2. if (a,b) and (a,c) are in f, then b=c.
    If a is an element of A, the unique element b that is an element of B for which (a,b) is an element of f is denoted by f(a)
    Follow Math Help Forum on Facebook and Google+

  8. #8
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by atrain313131 View Post
    Sorry about that - this is how it is written in the book:

    "Two functions are equal if they have the same domain, the same codomain, and "agree" on every element of the domain. More formally,

    Theorem: If f: A -> B and g: A -> B are functions, then f=g if and only if f(a) = g(a) for all a that are an element of A.

    (Hint: a function is a relation. Saying that two functions are equal is to say they are relations between the same pair of sets, and they include the same ordered pairs.)"
    Ah, well that's all you need then.

    write down what the ordered pairs of f look like. what about the ordered pairs of g? compare them. If they are the same, you're done.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by Jhevon View Post
    Ah, well that's all you need then.

    write down what the ordered pairs of f look like. what about the ordered pairs of g? compare them. If they are the same, you're done.
    of course, the "if and only if" means you'd have to prove both ways, so you might want to be careful on how you set it up. there are ways to avoid the whole having to do it both ways thing, since you're dealing with equality here, but in general, you want to be good and show both directions.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Word problem: Dealing with functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 19th 2011, 05:38 PM
  2. [SOLVED] Help proving by Mean value theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 17th 2010, 07:24 PM
  3. Need help with a proof dealing with functions
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 19th 2009, 08:37 AM
  4. Replies: 0
    Last Post: November 13th 2009, 06:41 AM
  5. Dealing with user-defined functions
    Posted in the Calculators Forum
    Replies: 0
    Last Post: October 27th 2008, 02:25 PM

Search Tags


/mathhelpforum @mathhelpforum