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?

Re: Bijective, surjective, injective functions

Quote:

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

Re: Bijective, surjective, injective functions

Quote:

Originally Posted by

**topsquark**

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

Re: Bijective, surjective, injective functions

Quote:

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*.)