One-to-One Function

  • #1
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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.
 
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  • #2
PeroK
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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 single domain 1 then it said to be a one-to-one function.

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.
 
  • #3
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0
Yes! Thank you
 

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