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 khổng lồ the same “B” (many-to-one is OK)

Injective means we won”t have two or more “A”s pointing to 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 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 và Surjective sầu 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 khổng lồ mean injective).

Bijective sầu functions have sầu an inverse!

If every “A” goes to a quality “B”, và 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 understvà 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 & 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” 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 lượt thích this:

*
A General Function

It CAN (possibly) have 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, and looks like this:

*
“Injective” (one-to-one)

In fact we can bởi a “Horizontal Line Test”:

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

(Note: Strictly Increasing (& Strictly Decreasing) functions are Injective, you might lượt thích 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 sầu function

Formal Definitions

OK, stvà by for more details about all this:

Injective

A function f is injective if và only if whenever f(x) = f(y), x = y.

Example: f(x) = x+5 from the phối of real numbers khổng lồ is an injective sầu 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 function because of this kind of thing:

f(2) = 4 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 lớn one B.

BUT if we made it from the set of naturalnumbers lớn then it is injective sầu, because:

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

So the domain name và cotên miền of each mix is important!

Surjective sầu (Also Called “Onto”)

A function f (from set A to B) is surjective if & only if for every y in B, there is at least one x in A such that f(x) = y, in other words f is surjective sầu if và only iff(A) = B.

In simple terms: every B has some A.

Example: The function f(x) = 2x from the phối of naturalnumbers to 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 thành viên in can be mapped lớn 3 by this function.

Bijective

A function f (from phối A lớn B) is bijective if, for every y in B, there is exactly one x in 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 và surjective.

Example: The function f(x) = x2 from the phối of positive realnumbers lớn positive sầu realnumbers is both injective and surjective sầu.Thus it is also bijective.

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

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