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Thread: Finding a Function

  1. #1
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    Finding a Function

    Let A= {a,b,c} and B= {1,2,3};$\displaystyle f : A \rightarrow B $ and $\displaystyle g: A \rightarrow B $ be a function defined by

    $\displaystyle f $
    $\displaystyle a \rightarrow 3 $
    $\displaystyle b \rightarrow 1 $
    $\displaystyle c \rightarrow 2 $

    $\displaystyle g $
    $\displaystyle a \rightarrow 1 $
    $\displaystyle b \rightarrow 3 $
    $\displaystyle c \rightarrow 1 $


    a) Find the function $\displaystyle F: B \rightarrow A $ such that $\displaystyle F \circ f = 1_A $. What is $\displaystyle f \circ F $?
    b) Find a function H: $\displaystyle A \rightarrow A $ such that $\displaystyle f \circ H = g$.

    for (a) am I suppose to find out which one it mapps to cause if so I dont undertstand and I have a list of problems like this.
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  2. #2
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    Quote Originally Posted by tigergirl View Post
    Let A= {a,b,c} and B= {1,2,3};$\displaystyle f : A \rightarrow B $ and $\displaystyle g: A \rightarrow B $ be a function defined by

    $\displaystyle f $
    $\displaystyle a \rightarrow 3 $
    $\displaystyle b \rightarrow 1 $
    $\displaystyle c \rightarrow 2 $

    $\displaystyle g $
    $\displaystyle a \rightarrow 1 $
    $\displaystyle b \rightarrow 3 $
    $\displaystyle c \rightarrow 1 $


    a) Find the function $\displaystyle F: B \rightarrow A $ such that $\displaystyle F \circ f = 1_A $. What is $\displaystyle f \circ F $?
    b) Find a function H: $\displaystyle A \rightarrow A $ such that $\displaystyle f \circ H = g$.

    for (a) am I suppose to find out which one it mapps to cause if so I dont undertstand and I have a list of problems like this.
    I would prefer drawing a picture - but let's try writing symbols instead.
    Try to think about what exactly $\displaystyle F \circ f = 1_A $ means
    You can rewrite it as the following 3 conditions:
    a=F(f(a))=F(3)
    b=F(f(b))=F(1)
    c=F(f(c))=F(2).
    Now we know the values of F(1),F(2),F(3) - the function F is fully determined, isn't it?

    Note that (if you draw a picture) F is the same as f, but with all arrows reversed.

    I think you could be able to do the second part in a similar way.
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