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Definition of injectivity

  1. Feb 17, 2009 #1
    I have a concern about the definition of injectivity:

    f:U->V; f is injective, for a,b in U

    1. a!=b implies f(a)!=f(b)
    2. f(a)=f(b) implies a=b

    Why isn't the definition:
    3. a!=b if and only if f(a)!=f(b)
    similarly,
    4. a=b if and only if f(a)=f(b)


    From 1, if a!=b implies f(a)!=f(b); consider a=b; certainly f(a)=f(b); so why isn't it that a!=b if and only if f(a)!=f(b).

    What is the logic behind the definition of injectivity?
     
  2. jcsd
  3. Feb 17, 2009 #2

    HallsofIvy

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    Since "if a= b then f(a)= f(b)" is part of the definition of "function", it not necessary to include it in the definition of "injective function".
     
    Last edited: Feb 19, 2009
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