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One-to-One Function

  1. Nov 11, 2014 #1
    Hey I was reading Susanna Discrete book and I came across her definition of One-to-One function:

    Let F be a function from a set X to a set Y. F is one-to-one (or injective) if, and only if, for all elements x1 and x2 in X,

    if F(x1 ) = F(x2 ),then x1 = x2 ,
    or, equivalently, if x1 ≠ x2 ,then F(x1) ≠ F(x2).

    Symbolically,
    F: X → Y is one-to-one ⇔ ∀x 1 ,x 2 ∈ X,if F(x1 ) = F(x2 ) then x1 = x2.

    But I am not sure if I fully understand the definition. Here is my interpretation of the definition:

    A function is said to be one-to-one if and only if,

    if f(x1) and f(x2) are the same then x1=x2 ,

    e.g if f(x1)=f(x2)=3, then
    x1 = x2 = 1

    Since, the co-domain 3 is being pointed by a two non-distinctive domain 1 then it said to be a one-to-one function.
     
    Last edited: Nov 11, 2014
  2. jcsd
  3. Nov 11, 2014 #2

    PeroK

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    Yes, I think you've got it. You can also think of what happens iff f is not 1-1:

    ##f \ \ is \ \ not \ \ 1-1 \ \ iff \ \ \exists x_1 \ne x_2 \ \ with \ \ f(x_1) = f(x_2)##

    That's a useful way to look at it as well.
     
  4. Nov 11, 2014 #3
    Yes! Thank you
     
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