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Math Help - Help on invertible function question

  1. #1
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    Help on invertible function question

    I like to know if anyone can tell me if my answer is correct.

    Suppose f is an invertible function from Y to Z and g is an invertible function from X to Y. Show that the inverse of the composition is f o g is given by (f o g)-1 = g-1 o f -1

    Answer:
    f –1(z) = y, g-1(y) =x
    (f o g)-1(y) = f -1 (g(y))-1 =x

    (g-1 o f -1)(y) = g-1(f(y))-1= z

    Therefore, (f o g)-1 = g-1 o f -1 are not equal. In other words, the commutative law does not hold for the composition function.

    (The f -1 and g-1 stands for the power of -1, sorry i don't have the button function to purt it correctly)
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  2. #2
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    \begin{array}{rcl}   {\left( {c,a} \right) \in \left( {f \circ g} \right)^{ - 1} } &  \Leftrightarrow  & {\left( {a,c} \right) \in f \circ g}  \\   {} &  \Rightarrow  & {\left( {\exists b} \right)\left[ {\left( {a,b} \right) \in g \wedge \left( {b,c} \right) \in f} \right]}  \\\end{array} \begin{array}{rcl}   {} &  \Rightarrow  & {\left( {c,b} \right) \in f^{ - 1}  \wedge \left( {b,a} \right) \in g^{ - 1} }  \\   {} &  \Rightarrow  & {\left( {c,a} \right) \in g^{ - 1}  \circ f^{ - 1} }  \\ \end{array}

    Now you can follow that and do the other half.
    Last edited by Plato; June 20th 2013 at 01:09 PM.
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  3. #3
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    i appreciate your help Plato, thanks a lot
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