One-to-One Function: Definition & Examples

[Nert]

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.

 

[PeroK]

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.

 

[Nert]

Yes! Thank you