Video Transcript
Choose the statement that describes
an injective function. Option (A) for every element in the
domain, there must be at least two corresponding elements in the range. Option (B) for every element in the
codomain, there must be a corresponding element in the domain. Option (C) no element in the range
can have more than one element in the domain corresponding to it. Option (D) for every element in the
domain, there is exactly one element corresponding to it in the range. Or is it option (E) for every
element in the range, there is more than one element corresponding to it in the
domain and vice versa?
In this question, we’re asked to
determine which of five given statements correctly describes an injective
function. And the easiest way to do this is
to start by recalling the definition of an injective function. We can recall that we call a
function an injective function if every element in the range of that function
corresponds to exactly one element of the domain. And if we check the five given
options, we can see none of the five given options match this word for word. So let’s instead go through each
option individually.
Let’s start with option (A). Option (A) tells us that for every
element in the domain, there must be at least two corresponding elements in the
range. But this presents a problem. This means we can input a value,
and we must get two different outputs. In other words, the statement in
option (A) does not represent a function. So in particular, it cannot
represent an injective function. All functions must have one output
per input. In other words, we can’t have
multiple elements in the range representing single elements in the domain.
Let’s now move on to option
(B). This tells us for every element in
the codomain, there must be a corresponding element in the domain. And at first, it might seem that
this is the correct answer. However, we need to be careful. We’re not told any information
about the number of elements. And the easiest way to see why this
does not necessarily represent an injective function is to see an example.
For example, let’s consider the
following function 𝑓 defined by this arrow diagram. The domain of this function is the
set containing one and two. And we have defined the codomain
and the range of this function just to be equal to zero since we’re going to have
this function only output zero. And it is worth noting the codomain
and range of a function do not need to be the same. The range is the set of all
possible outputs, but the codomain is the set containing the range. However, we can choose them to be
the same since we’re choosing our function 𝑓.
We can see that our function 𝑓
satisfies statement (B). To do this, we need to check that
every element of the codomain of our function has a corresponding element in the
domain. And we can just check this from the
given diagram. We have one element in the codomain
we need to check has a corresponding element in the domain. In fact, it has two: 𝑓 evaluated
at one is zero and 𝑓 evaluated at two is zero. However, using the same logic, we
can show that this function is not an injective function since injective functions
must have each element in the range corresponding to exactly one element of the
domain. So because there are two elements
which are mapped to zero in our function 𝑓, we know that it’s not an injective
function. So the statement in option (B)
can’t represent injective functions.
In fact, this represents a
different type of function called a surjective function. These are functions with the same
codomain and range. In other words, every element in
the codomain of the function must have a corresponding element in the domain. So every element in the codomain is
mapped onto. But this doesn’t help us answer the
question. So let’s now move on to option
(C).
Option (C) tells us that no element
in the range can have more than one element in the domain corresponding to it. Let’s once again sketch a function
diagram using arrows to try and determine what this property is telling us. Let’s start with the domain which
is the set of elements one, two, and three and a codomain which is the set one, two,
and three.
In statement (C), we’re told that
no element in the range can have more than one element in the domain corresponding
to it. And we can start by recalling the
range of a function is the set of all outputs of the function. In other words, it will be all of
the elements in the codomain which have an arrow pointing to them. So let’s say that we want one to be
in the range of our function. This means there must be an element
in the domain which corresponds to one.
We can choose any element in the
domain to be this value. So let’s say that 𝑓 evaluated at
one is one. This property now tells us that we
cannot have more than one element corresponding to one. So statement (C) tells us we now
cannot have 𝑓 evaluated at two is one. And now we can notice the
similarity with the definition of an injective function. Every element in the range
corresponds to exactly one element of the domain. This is just a slightly different
wording of the same statement.
This time, we’re being told for any
element in the range of our function, that means there is an arrow pointing to it in
the codomain, there is exactly one element in the domain pointing to it. We can’t have two arrows pointing
to the same element in the codomain. And so we’ve shown (C) is the
correct answer. However, for due diligence, let’s
also check options (D) and (E).
Option (D) tells us for every
element in the domain, there is exactly one element corresponding to it in the
range. We can once again consider this by
using an arrow diagram. This time, we’re told that every
element in the domain of this function corresponds to exactly one element in the
codomain. But we’re not told any information
about the uniqueness or number of arrows pointing to elements in the codomain.
For example, we could have that 𝑓
evaluated at one is one and 𝑓 evaluated at two is also one. We can see that every element in
the domain of this function has exactly one element corresponding to it in the
range. And of course, this is not an
injective function since we can see there are two elements which output one: 𝑓 of
one is one and 𝑓 of two is one. So option (D) does not represent an
injective function.
However, we can recognize this
definition as the definition of a function. This statement is just telling us
for every input value, we have exactly one output value. This is a function. But it’s not an injective function,
so let’s now move on to option (E).
Option (E) tells us for every
element in the range, there is more than one element corresponding to it in the
domain and vice versa. And although we could construct an
arrow diagram representing this relation, we can already use the previous
option. In this function, we can see every
element in the range of this function corresponds to more than one element in the
domain. And this is the first criteria of
option (E). However, we’ve already explained
that function 𝑓 cannot be an injective function. Just the first part of this
statement tells us that this function cannot be injective, so option (E) is not
correct.
Therefore, of the five given
options, only option (C) describes an injective function. It’s one where no elements in the
range can have more than one element in the domain corresponding to it.
