Surjective Là Gì

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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 AB 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 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 x in A such that f(x) = y, in other words f is surjective if & only iff(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 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 & 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)=4FunctionsSetsCommon Number SetsDomain, Range và CodomainSets Index