Surjective Là Gì

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It never has one “A” pointing khổng lồ 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 lớn 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).

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

But we can have sầu 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 together.

Think of it as a “perfect pairing” between the sets: every one has a partner và 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 an inverse!

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

Read Inverse Functions for more.

On A Graph

So let us see a few examples lớn understvà what is going on.

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

Let us have A on the x axis & 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” & 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 sầu Function” is stricter, và looks lượt thích this:

*
“Injective” (one-to-one)

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

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

(Note: Strictly Increasing (và Strictly Decreasing) functions are Injective sầu, 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, stand by for more details about all this:

Injective

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

Example: f(x) = x+5 from the mix of real numbers lớn 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 mix 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 phối of naturalnumbers lớn then it is injective, because:

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

So the domain & codomain name of each mix is important!

Surjective (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 if and 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 set of non-negative even numbers is a surjective function.

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

Bijective

A function f (from mix 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 sầu and surjective.

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Example: The function f(x) = x2 from the set of positive sầu realnumbers khổng lồ positive 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 because we could have sầu, for example, both

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