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Math Help - Characteristic Function on A

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    Characteristic Function on A

    Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5].

    a.)Sketch the graph of XAXB

    (which has X with an A subscript and X with B subscript, i suppose means XA composite XB)

    b.)Sketch the graph f|AIB


    Now i know how to graph XA and XB but im not sure what to do with a.) and i dont know what b.) is even asking.

    Any help or suggestions. Much appreciated.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Characteristic Function on A

    Quote Originally Posted by Aquameatwad View Post
    Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5].

    a.)Sketch the graph of XAXB

    (which has X with an A subscript and X with B subscript, i suppose means XA composite XB)

    b.)Sketch the graph f|AIB


    Now i know how to graph XA and XB but im not sure what to do with a.) and i dont know what b.) is even asking.

    Any help or suggestions. Much appreciated.
    Try to prove (or note) that in general that, in your notation, \chi_A\chi_B=\chi_{A\cap B}. I don't know what they're asking for b) either, what does the vertical line indicate? What are A,IB?
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    Re: Characteristic Function on A

    XAXB i suppose i should have written it as. XA composite XB. I'm not sure if its the intersection. IB i think is the Identity function: B-->B
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    Re: Characteristic Function on A

    Quote Originally Posted by Aquameatwad View Post
    Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5].

    a.)Sketch the graph of XAXB

    (which has X with an A subscript and X with B subscript, i suppose means XA composite XB)
    For any x, (\chi_A\circ\chi_B)(x)=\chi_A(\chi_B(x)). Now, \chi_B(x) returns either 0 or 1, so what's the value of \chi_A(\chi_B(x))?

    Quote Originally Posted by Aquameatwad View Post
    b.)Sketch the graph f|AIB
    Quote Originally Posted by Aquameatwad View Post
    IB i think is the Identity function: B-->B
    In this case, it looks like f|_A\circ I_B, i.e., the composition of the restriction of f on A and the identity function on B. So, (f|_A\circ I_B)(x)=f|_A(I_B(x))=f|_A(x)=2-3x. Since I_B is defined on (2, 5], the question now is only to determine for which x\in(2,5] the function application f|_A(x) is defined.
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    Re: Characteristic Function on A

    Well for part a. would (2,5] be on 1? So would it just be as if we were graphing just XB? Cause if i put XB inside XA which is what the composite is wouldnt it just be the intersection of XA and XB?

    Its not making sense to my pea brain.
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    Re: Characteristic Function on A

    Quote Originally Posted by emakarov View Post
    For any x, (\chi_A\circ\chi_B)(x)=\chi_A(\chi_B(x)). Now, \chi_B(x) returns either 0 or 1, so what's the value of \chi_A(\chi_B(x))?
    If I understand the definitions correctly,

    \chi_A(x)=\begin{cases}1&x\in[-1,3)\\0&\mbox{otherwise}\end{cases}

    \chi_B(x)=\begin{cases}1&x\in(2,5]\\0&\mbox{otherwise}\end{cases}

    Case 1: x\in(2,5]. Then \chi_B(x)=1, so \chi_A(\chi_B(x))=\dots

    Case 2: x\notin(2,5]. Then \chi_B(x)=0, so \chi_A(\chi_B(x))=\dots

    As an aside note, one of the confusing things about \chi_A(\chi_B(x)) is the different meaning given to real numbers. The output of \chi_B(x) is 0 or 1, yet it is interpreted as any other real number and is fed to \chi_A. In programming, it would be clearer to make \chi_A and \chi_B return Boolean values True and False instead of numbers 1 and 0. Then \chi_A(\chi_B(x)) would be ill-typed.
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    Re: Characteristic Function on A

    The syntax and math jargon of this question is probably whats making this difficult to me.

    So what does it mean for instance in case1: XA(1) and case 2: XA(0).?

    For case 1: Would i graph the line then at y=1, x=[ -1,3) for XA=1. ??
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    Re: Characteristic Function on A

    Before sketching anything, you need to learn to compute compositions of functions. So, what are \chi_A(1) and \chi_A(0), and therefore, what is \chi_A(\chi_B(x)) for all x?
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    Re: Characteristic Function on A

    i understand the composition of functions and relations , but A and B are just sets arent they? This isnt making any sense. The characteristic functions aren't given much explanation in my textbook. So i have no idea what there purpose even is. It's just a review problem our teacher gave us and I'm just trying to graph it.
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