# 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":

Let's look at that more closely:

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 we won't have two or more "A"s pointing to the same "B".

So **many-to-one is NOT OK** (which is OK for a general function).

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

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

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.

Think of it as a "perfect pairing" between the sets: every one has a partner and no one is left out.

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).

Bijective functions have an **inverse**!

If every "A" goes to a unique "B", and every "B" has a matching "A" then we can go back and forwards without being led astray.

Read Inverse Functions for more.

## On A Graph

So let us see a few examples to understand what is going on.

When **A** and **B** are subsets of the Real Numbers we can graph the relationship.

Let us have **A** on the x axis and **B** on y, and look at our first example:

This is **not a function** because we have an **A** with many **B**. It is like saying f(x) = 2 * or* 4

It fails the "Vertical Line Test" and so is not a function. But is still a valid relationship, so don't get angry with it.

Now, a general function can be like this:

A General Function

It CAN (possibly) have a **B** with many **A**. For example sine, cosine, etc are like that. Perfectly valid functions.

But an "**Injective Function**" is stricter, and looks like this:

"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)*

So:

- If it passes the
**vertical line test**it is a function - If it also passes the
**horizontal line test**it is an injective function

## Formal Definitions

OK, stand by for more 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.

Is it true that whenever ** f(x) = f(y)**,

**?**

*x = y*Imagine x=3, then:

- f(x) = 8

Now I say that f(y) = 8, what is the value of y? It can only be 3, so x=y

**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**.

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

So the domain and codomain of each set is important!

### 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*In simple terms: every B has some A.

**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, no member 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 both injective and surjective. Thus it is also

**bijective**.

But the same function from the set of all real numbers is not bijective because we could have, for example, both

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