Videos > Surjective (onto) and injective (one-to-one) functions
Surjective (onto) and injective (one-to-one) functions Video Lecture
FAQs on Surjective (onto) and injective (one-to-one) functions Video Lecture
| 1. What is a surjective function? |  |
Ans. A surjective function, also known as an onto function, is a function in which every element in the range is mapped to by at least one element in the domain. In other words, for every y in the range, there exists an x in the domain such that f(x) = y.
| 2. What is an injective function? | |
Ans. An injective function, also known as a one-to-one function, is a function in which every element in the domain is mapped to a unique element in the range. In other words, for every x1 and x2 in the domain, if f(x1) = f(x2), then x1 = x2.
| 3. How can we determine if a function is surjective? | |
Ans. To determine if a function is surjective, we need to check if every element in the range has a pre-image in the domain. One way to do this is to analyze the graph of the function and see if it passes the horizontal line test. If every horizontal line intersects the graph at least once, then the function is surjective.
| 4. How can we determine if a function is injective? | |
Ans. To determine if a function is injective, we need to check if every element in the range corresponds to a unique element in the domain. One way to do this is to check if the function is strictly increasing or strictly decreasing. If there are no two distinct elements in the domain that map to the same element in the range, then the function is injective.
| 5. Can a function be both surjective and injective? | |
Ans. Yes, a function can be both surjective and injective. Such a function is called a bijective function. In a bijective function, every element in the domain is mapped to a unique element in the range, and every element in the range is mapped to by exactly one element in the domain. Bijective functions have an inverse function, which allows us to "undo" the mapping and retrieve the original values.
Text Transcript from Video
In this video I want to
introduce you to some
terminology that will be useful
in our discussion of
functions and invertibility.
And this is, in general,
terminology that you'll
probably see in your
mathematical careers.
So let's say I have a function
f, and it is a mapping from
the set x to the set y.
We've drawn this diagram many
times, but it never hurts to
draw it again.
So that is my set
x or my domain.
And then this is the set y over
here, or the co-domain.
Remember the co-domain is the
set that you're mapping to.
You don't necessarily have to
map to every element of the
set, or none of the elements
of the set.
This is just all of the
elements, the set that you
might map elements in
your co-domain to.
So let's see.
If I have some element there, f
will map it to some element
in y in my co-domain.
So the first idea, or term, I
want to introduce you to, is
the idea of a function
being surjective.
And sometimes this
is called onto.
And a function is surjective or
onto, if for every element
in your co-domain-- so let me
write it this way, if for
every, let's say y, that is a
member of my co-domain, there
exists-- that's the little
shorthand notation for exists
--there exists at least
one x that's a
member of x, such that.
And I can write such
that, like that.
Actually, let me just
write the word out.
Such that f of x
is equal to y.
So it's essentially saying, you
can pick any y here, and
every y here is being mapped
to by at least one of
the x's over here.
So, for example, actually let
me draw a simpler example
instead of drawing
these blurbs.
Let's say that I have
a set y that
literally looks like this.
Let's say that a set y-- I'll
draw it very --and let's say
it has four elements.
It has the elements
a, b, c, and d.
This is my set y right there.
And let's say my set
x looks like that.
And let's say it has the
elements 1, 2, 3, and 4.
Now, in order for my function f
to be surjective or onto, it
means that every one of these
guys have to be able
to be mapped to.
So what does that mean?
If every one of these
guys, let me
just draw some examples.
Let's say that this
guy maps to that.
Let's say that this
guy maps to that.
Let's say that this
guy maps to that.
And let's say, let me draw a
fifth one right here, let's
say that both of these guys
right here map to d.
So f of 4 is d and
f of 5 is d.
This is an example of a
surjective function.
So these are the mappings
of f right here.
This function right here
is onto or surjective.
Why is that?
Because every element here
is being mapped to.
Now, let me give you an example
of a function that is
not surjective.
Let me add some more
elements to y.
Let's say element y has another
element here called e.
Now, all of a sudden, this
is not surjective.
And why is that?
Because there's some element
in y that is not
being mapped to.
If I tell you that f is a
surjective function, it means
if you take, essentially, if you
map all of these values,
everything here is being mapped
to by at least one
element here.
So you could have it, everything
could be kind of a
one-to-one mapping.
And I'll define that a little
bit better in the future.
So it could just be like
that, and like that.
And you could even have, it's
at least one, so you could
even have two things in here
mapping to one thing in here.
But the main requirement
is that everything here
does get mapped to.
Another way to think about it,
is that if you take the image.
So surjective function--
let me write this here.
Let me write it this way --so if
I say that f is surjective
or onto, these are equivalent
terms, that means that the
image of f.
Remember the image was, all
of the values that f
actually maps to.
So that means that the image
of f is equal to y.
Now, we learned before, that
your image doesn't have to
equal your co-domain.
But if you have a surjective
or an onto function, your
image is going to equal
your co-domain.
Everything in your co-domain
gets mapped to.
Actually, another word
for image is range.
You could also say that your
range of f is equal to y.
Remember the difference-- and
I drew this distinction when
we first talked about functions
--the distinction
between a co-domain and a range,
a co-domain is the set
that you can map to.
You don't have to map
to everything.
The range is a subset of
your co-domain that you
actually do map to.
If you were to evaluate the
function at all of these
points, the points that you
actually map to is your range.
And that's also called
your image.
And the word image
is used more in a
linear algebra context.
But if your image or your
range is equal to your
co-domain, if everything in your
co-domain does get mapped
to, then you're dealing
with a surjective
function or an onto function.
Now, the next term I want to
introduce you to is the idea
of an injective function.
And this is sometimes called
a one-to-one function.
So let me draw my domain
and co-domain again.
So let's say that that
is my domain
and this is my co-domain.
So this is x and this is y.
If I say that f is injective
or one-to-one, that implies
that for every value that is
mapped to-- so let me write it
this way --for every value that
is mapped to-- so let's
say, I'll say it a couple of
different ways --there is at
most one x that maps to it.
Or another way to say it is that
for any y that's a member
of y-- let me write it this
way --for any y that is a
member y, there is at most one--
let me write most in
capital --at most one x, such
that f of x is equal to y.
There might be no x's
that map to it.
So for example, you could have
a little member of y right
here that just never
gets mapped to.
Everyone else in y gets mapped
to, but that guy
never gets mapped to.
So this would be a case
where we don't have
a surjective function.
This is not onto because this
guy, he's a member of the
co-domain, but he's not
a member of the
image or the range.
He doesn't get mapped to.
But this would still be an
injective function as long as
every x gets mapped
to a unique y.
Now, how can a function not be
injective or one-to-one?
And I think you get the idea
when someone says one-to-one.
Well, if two x's here get mapped
to the same y, or three
get mapped to the same y, this
would mean that we're not
dealing with an injective or
a one-to-one function.
So that's all it means.
Let me draw another
example here.
Let's actually go back to
this example right here.
When I added this e here, we
said this is not surjective
anymore because every one
of these guys is not
being mapped to.
Is this an injective function?
Well, no, because I have f of 5
and f of 4 both mapped to d.
So this is what breaks its
one-to-one-ness or its
injectiveness.
This is what breaks it's
surjectiveness.
Now if I wanted to make this a
surjective and an injective
function, I would delete that
mapping and I would change f
of 5 to be e.
Now everything is one-to-one.
I don't have the mapping from
two elements of x, going to
the same element of y anymore.
And everything in y now
gets mapped to.
So this is both onto
and one-to-one.
introduce you to some
terminology that will be useful
in our discussion of
functions and invertibility.
And this is, in general,
terminology that you'll
probably see in your
mathematical careers.
So let's say I have a function
f, and it is a mapping from
the set x to the set y.
We've drawn this diagram many
times, but it never hurts to
draw it again.
So that is my set
x or my domain.
And then this is the set y over
here, or the co-domain.
Remember the co-domain is the
set that you're mapping to.
You don't necessarily have to
map to every element of the
set, or none of the elements
of the set.
This is just all of the
elements, the set that you
might map elements in
your co-domain to.
So let's see.
If I have some element there, f
will map it to some element
in y in my co-domain.
So the first idea, or term, I
want to introduce you to, is
the idea of a function
being surjective.
And sometimes this
is called onto.
And a function is surjective or
onto, if for every element
in your co-domain-- so let me
write it this way, if for
every, let's say y, that is a
member of my co-domain, there
exists-- that's the little
shorthand notation for exists
--there exists at least
one x that's a
member of x, such that.
And I can write such
that, like that.
Actually, let me just
write the word out.
Such that f of x
is equal to y.
So it's essentially saying, you
can pick any y here, and
every y here is being mapped
to by at least one of
the x's over here.
So, for example, actually let
me draw a simpler example
instead of drawing
these blurbs.
Let's say that I have
a set y that
literally looks like this.
Let's say that a set y-- I'll
draw it very --and let's say
it has four elements.
It has the elements
a, b, c, and d.
This is my set y right there.
And let's say my set
x looks like that.
And let's say it has the
elements 1, 2, 3, and 4.
Now, in order for my function f
to be surjective or onto, it
means that every one of these
guys have to be able
to be mapped to.
So what does that mean?
If every one of these
guys, let me
just draw some examples.
Let's say that this
guy maps to that.
Let's say that this
guy maps to that.
Let's say that this
guy maps to that.
And let's say, let me draw a
fifth one right here, let's
say that both of these guys
right here map to d.
So f of 4 is d and
f of 5 is d.
This is an example of a
surjective function.
So these are the mappings
of f right here.
This function right here
is onto or surjective.
Why is that?
Because every element here
is being mapped to.
Now, let me give you an example
of a function that is
not surjective.
Let me add some more
elements to y.
Let's say element y has another
element here called e.
Now, all of a sudden, this
is not surjective.
And why is that?
Because there's some element
in y that is not
being mapped to.
If I tell you that f is a
surjective function, it means
if you take, essentially, if you
map all of these values,
everything here is being mapped
to by at least one
element here.
So you could have it, everything
could be kind of a
one-to-one mapping.
And I'll define that a little
bit better in the future.
So it could just be like
that, and like that.
And you could even have, it's
at least one, so you could
even have two things in here
mapping to one thing in here.
But the main requirement
is that everything here
does get mapped to.
Another way to think about it,
is that if you take the image.
So surjective function--
let me write this here.
Let me write it this way --so if
I say that f is surjective
or onto, these are equivalent
terms, that means that the
image of f.
Remember the image was, all
of the values that f
actually maps to.
So that means that the image
of f is equal to y.
Now, we learned before, that
your image doesn't have to
equal your co-domain.
But if you have a surjective
or an onto function, your
image is going to equal
your co-domain.
Everything in your co-domain
gets mapped to.
Actually, another word
for image is range.
You could also say that your
range of f is equal to y.
Remember the difference-- and
I drew this distinction when
we first talked about functions
--the distinction
between a co-domain and a range,
a co-domain is the set
that you can map to.
You don't have to map
to everything.
The range is a subset of
your co-domain that you
actually do map to.
If you were to evaluate the
function at all of these
points, the points that you
actually map to is your range.
And that's also called
your image.
And the word image
is used more in a
linear algebra context.
But if your image or your
range is equal to your
co-domain, if everything in your
co-domain does get mapped
to, then you're dealing
with a surjective
function or an onto function.
Now, the next term I want to
introduce you to is the idea
of an injective function.
And this is sometimes called
a one-to-one function.
So let me draw my domain
and co-domain again.
So let's say that that
is my domain
and this is my co-domain.
So this is x and this is y.
If I say that f is injective
or one-to-one, that implies
that for every value that is
mapped to-- so let me write it
this way --for every value that
is mapped to-- so let's
say, I'll say it a couple of
different ways --there is at
most one x that maps to it.
Or another way to say it is that
for any y that's a member
of y-- let me write it this
way --for any y that is a
member y, there is at most one--
let me write most in
capital --at most one x, such
that f of x is equal to y.
There might be no x's
that map to it.
So for example, you could have
a little member of y right
here that just never
gets mapped to.
Everyone else in y gets mapped
to, but that guy
never gets mapped to.
So this would be a case
where we don't have
a surjective function.
This is not onto because this
guy, he's a member of the
co-domain, but he's not
a member of the
image or the range.
He doesn't get mapped to.
But this would still be an
injective function as long as
every x gets mapped
to a unique y.
Now, how can a function not be
injective or one-to-one?
And I think you get the idea
when someone says one-to-one.
Well, if two x's here get mapped
to the same y, or three
get mapped to the same y, this
would mean that we're not
dealing with an injective or
a one-to-one function.
So that's all it means.
Let me draw another
example here.
Let's actually go back to
this example right here.
When I added this e here, we
said this is not surjective
anymore because every one
of these guys is not
being mapped to.
Is this an injective function?
Well, no, because I have f of 5
and f of 4 both mapped to d.
So this is what breaks its
one-to-one-ness or its
injectiveness.
This is what breaks it's
surjectiveness.
Now if I wanted to make this a
surjective and an injective
function, I would delete that
mapping and I would change f
of 5 to be e.
Now everything is one-to-one.
I don't have the mapping from
two elements of x, going to
the same element of y anymore.
And everything in y now
gets mapped to.
So this is both onto
and one-to-one.
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Oct 06, 2025
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