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 two or more “A”s pointing lớn the same “B”.
So many-to-one is NOT OK (which is OK for a general function).
Đang xem: Surjective là gì
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 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 to lớn mean injective).
Bijective functions have an inverse!
If every “A” goes to lớn a quality “B”, & every “B” has a matching “A” then we can go back & 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 & B are subsets of the Real Numbers we can graph the relationship.
Let us have A on the x axis và 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” & 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:
It CAN (possibly) have a B with many A. For example sine, cosine, etc are lượt thích that. Perfectly valid functions.
But an “Injective Function” is stricter, và looks lượt thích this:
In fact we can bởi vì 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 lượt thích to read about them for more details)
If it passes the vertical line test it is a function If it also passes the horizontal line test it is an injective function
OK, stand by for more details about all this:
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 khổng lồ 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 mix of real numbers lớn is not an injective function because of this kind of thing:
f(2) = 4 and 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 mix 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 of each set 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 và only iff(A) = B.
In simple terms: every B has some A.
Example: The function f(x) = 2x from the mix of naturalnumbers khổng lồ the mix of non-negative even numbers is a surjective function.
BUT f(x) = 2x from the set of naturalnumbers khổng lồ is not surjective, because, for example, no member in can be mapped lớn 3 by this function.
A function f (from mix A to 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 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 mix of positive realnumbers to positive realnumbers is both injective và 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 & f(-2)=4FunctionsSetsCommon Number SetsDomain, Range and CodomainSets Index