# Thread: Bijective, surjective, injective functions

1. ## Bijective, surjective, injective functions

A General Function points from each member of "A" to a member of "B".

To be a function you never have one "A" pointing to more than one "B", so one-to-many is not OK in a function (as you would have something like "f(x) = 7 or 9")

Injective, Surjective and Bijective

What about a circle? Is that not a function? If not, why?

2. ## Re: Bijective, surjective, injective functions

Originally Posted by Paze
A General Function points from each member of "A" to a member of "B".

To be a function you never have one "A" pointing to more than one "B", so one-to-many is not OK in a function (as you would have something like "f(x) = 7 or 9")

Injective, Surjective and Bijective

What about a circle? Is that not a function? If not, why?
Use the vertical line test.

-Dan

3. ## Re: Bijective, surjective, injective functions

Originally Posted by topsquark
Use the vertical line test.

-Dan
Thank you. What is the difference between the vertical and the horizontal line test?

4. ## Re: Bijective, surjective, injective functions

Originally Posted by Paze
Thank you. What is the difference between the vertical and the horizontal line test?
Learn the actual definition of function.
The statement that $\displaystyle f:A\to B$ is a function from A to B means:
$\displaystyle \\ \bullet~f\subset A\times B \\ \bullet~A=\text{Dom}(f)\\ \bullet~\text{no two pairs in }f\text{ have the same first term.}$ That last bullet-point is the vertical line test.

Here are some special properties of some functions.
If $\displaystyle \text{Img}(f)=B$ then the function is surjective.

If $\displaystyle \text{no two pairs in }f\text{ have the same second term.}$ then the function is injective.( that is also known as the horizontal line test.)