# Injective, Surjective and Bijective

"Injective, Surjective and Bijective" tells us about how a function behaves.

A function is a way of matching the members of a set "A" **to** a set "B":

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

It **never** has one "A" pointing to more than one "B", so **one-to-many is not OK** in a function (so something like "f(x) = 7 * or* 9" is not allowed)

But more than one "A" can point to the same "B" (**many-to-one is OK**)

**Injective** means that every member of "A" has **its own unique** matching member in "B".

As it is also a function** one-to-many is not OK**

And we won't have two "A"s pointing to the same "B", so **many-to-one is NOT OK**.

But we can have a "B" without a matching "A"

Injective functions can be **reversed**!

If "A" goes to a unique "B" then given that "B" value we can go back again to "A" (this does not work when two or more "A"s pointed to one "B" like in the "General Function")

Read Inverse Functions for more.

Injective is also called "**One-to-One**"

**Surjective** means that every "B" has **at least one** matching "A" (maybe more than one).

There won't be a "B" left out.

**Bijective** means both Injective and Surjective together.

So there is a perfect "**one-to-one correspondence**" between the members of the sets.

(But don't get that confused with the term "One-to-One" used to mean injective).

## On The Graph

Let's see on a graph what a "**General Function**" and a "**Injective Function**" looks like:

General Function | "Injective" (one-to-one) |

In fact we can do a "Horizontal Line Test":

To be **Injective**, a Horizontal Line should never intersect the curve at 2 or more points.

*(Note: Strictly Increasing (and Strictly Decreasing) functions are Injective, you might like to read about them for more details)*

## Formal Definitions

OK, stand by for some details about all this:

### Injective

A function ** f** is

**injective**if and only if whenever

**,**

*f(x) = f(y)***.**

*x = y***Example:** ** f(x) = x+5** from the set of real numbers to is an injective function.

This function can be easily reversed. for example:

**f(3) = 8**

Given 8 we can go back to 3

**Example:** ** f(x) = x^{2}** from the set of real numbers to is

**not**an injective function because of this kind of thing:

and*f*(*2*) = 4*f*(*-2*) = 4

This is against the definition ** f(x) = f(y)**,

**, because**

*x = y*

**f(2) = f(-2) but 2 ≠ -2**In other words there are **two** values of "A" that point to one "B", and this function could not be reversed (given the value "4" ... what produced it?)

BUT if we made it from the set of natural numbers to then it is injective, because:

*f*(*2*) = 4- there is no f(-2), because -2 is not a natural number

### Surjective (Also Called "Onto")

A function ** f** (from set

*to*

**A***) is*

**B****surjective**if and only if for every

**in**

*y**, there is at least one*

**B****in**

*x**such that*

**A***f*(

*x*) =

*y*,

**in other words**

**is surjective if and only if**

*f***.**

*f(A) = B*So, every element of the range corresponds to at least one member of the domain.

**Example:** The function ** f(x) = 2x** from the set of natural
numbers to the set of non-negative

**even**numbers is a

**surjective**function.

BUT ** f(x) = 2x** from the set of natural
numbers to is

**not surjective**, because, for example, nothing in can be mapped to

**by this function.**

*3*

### Bijective

A function ** f** (from set

*to*

**A***) is*

**B****bijective**if, for every

**in**

*y**, there is exactly one*

**B****in**

*x**such that*

**A***f*(

*x*) =

*y*

Alternatively, ** f** is bijective if it is a

**one-to-one correspondence**between those sets, in other words both

**injective and surjective.**

**Example:** The function ** f(x) = x^{2}** from the set of positive real
numbers to positive real
numbers is injective and surjective.
Thus it is also

**bijective**.

But not from the set of real numbers because we could have, for example, both

*f*(*2*)=4 and*f*(*-2*)=4