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I Injectivity equivalent to having a left inverse

  1. Jul 18, 2017 #1
    I know that one can easily prove the result that a function is injective if and only if that function has a left inverse. But is there intuitive reason for this? Same goes for the fact that having a right inverse is equivalent to being surjective. Why are the properties of injectivity and sujectivity related to inverses?
     
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  3. Jul 19, 2017 #2

    andrewkirk

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    When a left inverse is applied to a function, as in ##f^{-1}\circ f(x)##, it 'undoes' the effect of the function because, under the right-to-left rule for function composition, ##f^{-1}## is applied after ##f##. If the function was not injective ##f^{-1}## could not map back to the original ##x## because there would be more than one possibility.

    When a right inverse is applied, as in ##f\circ f^{-1}(y)##, the right inverse ##f^{-1}## will not be defined on all elements of its domain - which is the range of ##f## - if ##f## is surjective.
     
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