Welcome to a lesson on image
and inverse image.
When discussing functions, we have notation
for talking about an element of the domain, say x,
and its corresponding element in the codomain,
we write f of x, which is the image of x.
Sometimes we'll want to talk about
all the elements that are images
of some subset of the domain.
It would also be nice to start with some element
of the codomain, say y,
and talk about which element or elements, if any,
from the domain it is the image of.
We could write those x in the domain
such that f of x equals y,
but this is a lot of writing
and therefore, we use special notation.
To address the first situation,
what we are after is a way to describe
the set of images of elements in some subset
of the domain.
Suppose f maps x to y,
meaning x is the domain and y is the codomain,
where big A is a subset of big X,
meaning big A is a subset of the domain, possibly all of it.
We will use the notation f of big A
to denote the image of A under f,
namely the set of elements in Y
that are the image of elements from the set A.
That is, f of big A is equal to the set
containing f of little A which are elements of big Y,
where again, big Y is the codomain
such that little A is an element of big A
where big A again is a subset of the domain.
Let's look at another example.
Below, we have the domain and the codomain
for the function given using two-line notation.
On the right, we have the image of the set of elements
one, two, and three,
which is equal to the set containing the elements
that are the images of one, two and three from the domain
which is equal to the set with elements 11 and 15.
Again, the set with elements 11 and 15
is the set of the images of the subset of the domain,
which is the set with elements one, two and three.
Next, we have the image of the set with elements one and two
equals the set with the element of 11.
This indicates that 11 is the image
of the elements of one and two from the domain
or the set with elements one and two
is a subset of the domain.
And then finally, we have the image
of the set with the element of five
equals a set with the element of 17.
Again, this is equal to the set
with the element of 17
because 17 is the image of the element
of five from the domain.
We can do this in the other direction as well.
We might ask which elements of the domain
get mapped to a particular set in the codomain.
So again, if f maps x to y
and we let big B be a subset of big Y,
meaning big B is a subset of the codomain,
then we'll use the notation shown here
for the inverse image of B under f
namely the set of elements in x the domain,
whose image are elements in B.
In other words, the inverse image of B under f
is equal to the set of x
where x is an element of the domain
such that f of little x is an element of b
where b is a subset of the codomain.
Often we are interested in the elements
whose image is a particular element y
in the codomain.
The notation above works,
where this is the set of all elements in the domain
that f sends to y.
It makes sense to think of this as a set.
There might not be anything sent to y,
if y is not in the range,
in which case, the inverse image of the set containing y
under f is equal to the empty set
or f might send multiple elements to y
if f is not injective.
As a notational convenience,
the curly brackets around the y can be dropped
and we can use this notation instead,
but I'm going to go ahead and keep using
the curly brackets for a set notation.
And I also look at the same function again
and look at some inverse images.
So again, we have the same function, same domain,
same codomain, but now in the right,
we have the inverse image of the set with the elements
11 and 15 is equal to
the set containing all the elements from the domain
in which 11 and 15 are the images.
Notice how this is a set containing the elements
of one, two, three and four.
Below, we have the inverse image of the set
with the element of 17
which is equal to the set with the elements from the domain
where 17 is the image.
And notice in this case, it's only the element of five
from the domain, which gives us the set
with the element of five.
Let's look at one more example.
We have a function f that maps a set
with elements one, two, three, four, five, six
to the set with elements A, B, C and D
given by the function shown below in two-line notation.
So first, we're asked to find
the image of the set with elements one, two, and three
under f, this is equal to the set
containing the elements that are images
of the elements one, two, and three from the domain.
So we're looking at the function and two-line notation,
notice one maps to a, two maps to a, and three maps to b
and therefore, the image of the set with elements
one, two and three under f
is equal to the set with elements a and b.
And this is because a and b are the elements
in the codomain to which f sends one, two and three.
Next, we have the inverse image
of the set with elements a and b under f.
This is equal to the set of elements from the domain
in which a and b are the images.
And notice the elements of one, two,
three, four, and five from the domain
do have images of a or b
and therefore, the inverse image of the set
with elements a and b under f
is equal to the set with the elements
of one, two, three, four and five.
And then finally, we have the inverse image
of the set with the element of d under f
which is equal to the set with the elements
from the domain where d is the image.
And notice in this case, d does not appear in the range
and therefore, no elements from the domain
have an image of d
and therefore, the inverse image of the set containing d
under f is equal to the empty set.
Again, d is not in the range of f.
I hope you found this helpful.