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Math Help - Finding a Function

  1. #1
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    Finding a Function

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

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

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


    a) Find the function  F: B \rightarrow A such that  F \circ f = 1_A . What is  f \circ F ?
    b) Find a function H:  A \rightarrow A such that  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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    Bratislava
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    Quote Originally Posted by tigergirl View Post
    Let A= {a,b,c} and B= {1,2,3};  f : A \rightarrow B and  g: A \rightarrow B be a function defined by

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

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


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