# Surjective Là Gì

It **never** has one “A” pointing to more than one “B”, so **one-to-many is not OK** in a function (so something lượt thích “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 sầu two or more “A”s pointing lớn the same “B”.

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

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

But we can have a “B” without a matching “A”

Injective sầu 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 & 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 sầu an **inverse**!

If every “A” goes to a chất lượng “B”, and every “B” has a matching “A” then we can go baông chồng and forwards without being led astray.

Read Inverse Functions for more.

## On A Graph

So let us see a few examples to lớn underst& what is going on.

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

Let us have **A** on the x axis and **B** on y, & 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” và so is not a function. But is still a valid relationship, so don”t get angry with it.

Now, a general function can be lượt thích this:

**A General Function**

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

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But an “**Injective Function**” is stricter, và looks lượt thích this:

**“Injective” (one-to-one)**

In fact we can vày 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 sầu, you might like lớn 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, st& 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 phối of real numbers to lớn 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) = x2** from the set of real numbers khổng lồ is **not** an injective sầu function because of this kind of thing:

**f(2) = 4 **và **f(-2) = 4**

This is against the definition **f(x) = f(y)**, ** x = y**, because **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 naturalnumbers to then it is injective sầu, because:

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

So the tên miền and codomain name of each phối is important!

### Surjective sầu (Also Called “Onto”)

A function **f** (from mix **A** to **B**) is **surjective** if and only if for every ** y** in

**B**, there is at least one

**in**

*x***A**such that

*f*(

*x*) =

*y*,

**in other words**

**f**is surjective if & only if

**f(A) = B**.

In simple terms: every B has some A.

**Example:** The function **f(x) = 2x** from the phối of naturalnumbers lớn the mix of non-negative **even** numbers is a **surjective** function.

BUT **f(x) = 2x** from the phối of naturalnumbers to is **not surjective**, because, for example, no member in can be mapped to lớn **3** by this function.

### Bijective

A function ** f** (from set

**A**khổng lồ

**B**) is

**bijective**if, for every

**in**

*y***B**, there is exactly one

**in**

*x***A**such that

*f*(

*x*) =

*y*

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

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

**injective & surjective.**

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**Example:** The function **f(x) = x2** from the set of positive sầu realnumbers to positive sầu realnumbers is both injective sầu & surjective.Thus it is also **bijective**.

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

**f(2)=4 & ****f(-2)=4**FunctionsSetsCommon Number SetsDomain, Range và CodomainSets Index