Video Transcript
Which of the following relations
represents a function from 𝑋 to 𝑌?
And then we have four possible
relations given by mapping diagrams. Now remember, a mapping diagram
takes elements from one set 𝑋 and maps them onto elements in a second set, in this
case, 𝑌. Then, a function will always be an
example of a relation, but we cannot say the reverse is necessarily true. All relations will not necessarily
be functions. And this is because a function must
map each element from the input, set 𝑋, onto exactly one element of the output, set
𝑌. So to establish which of the
relations represents a function from 𝑋 to 𝑌, we need to identify which of the
relations maps one element from set 𝑋 onto exactly one in set 𝑌.
Let’s begin by considering our
first relation. This relation maps negative two
onto negative four in set 𝑌. Similarly, it takes the element
zero from set 𝑋 and maps it onto one element in set 𝑌. Finally, the element one is mapped
onto exactly one element in set 𝑌, negative four. Since each element in set 𝑋 is
mapped onto only one element in set 𝑌, relation one must be a function. For completeness, let’s double
check (R2), (R3), and (R4).
Let’s first look at (R2). If we look carefully, we see that
this element negative two in set 𝑋 has two arrows coming from it. It is mapped onto negative four in
set 𝑌 and negative two. Since this element negative two
maps onto two elements in the output, we know that (R2), the second relation, cannot
be a function. Similarly, consider element one in
relation three. This is mapped onto the element
negative nine and the element negative two in set 𝑌. Since this element does not map
onto exactly one element in set 𝑌, we disregard (R3). This relation does not represent a
function. And finally, in our fourth
relation, the element negative two maps onto both the element negative nine and the
element negative two in set 𝑌.
It is not one to one. This element maps onto two elements
of the output, so we disregard relation four. And so (R1) relation one must
represent a function from 𝑋 to 𝑌.
