Video Transcript
If the set 𝑋 is equal to eight, 𝑌
is equal to eight, three, and 𝑍 is equal to nine, four, five, find the union of the
Cartesian products of 𝑋 and 𝑌 and 𝑌 and 𝑍.
So to solve this problem, what we
need to do is first find the products of 𝑋 and 𝑌 and 𝑌 and 𝑍, and then find
their union. So I’m gonna start with 𝑋
multiplied by 𝑌, or the product of 𝑋 and 𝑌. So to find the Cartesian product,
so we multiply 𝑋 and 𝑌, what we need to do is write down a set of ordered pairs
with the coordinates 𝑋, 𝑌 that come from 𝑋 being an element of 𝑋 and 𝑌 being an
element of 𝑌.
So as you could see, I’ve written
this definition down and I’ve used some set notations. I’ve got this giant ∈. What this means is an element
of. So what this means in real terms is
we’re gonna take the values from 𝑋 and 𝑌 and create ordered pairs of coordinates
from them. So therefore, we’re gonna have a
set of coordinates whose 𝑋-values are going to be eight and 𝑌-values are gonna be
three or eight. So therefore, the Cartesian product
of 𝑋 and 𝑌 is going to be eight, three and eight, eight because they are only
possible pairs of products. Because we’ve got the eight from
set 𝑋 with the three from set 𝑌 and then the eight from set 𝑋 with the eight from
set 𝑌.
So now, we’re gonna move on to 𝑌
and 𝑍, so 𝑌 multiplied by 𝑍. So with the Cartesian product of
𝑌 and 𝑍, we’re gonna have the coordinates 𝑌, 𝑍, where 𝑌 is gonna be either three
or eight and 𝑍 is gonna be either four, five, or nine. So therefore, our set of possible
results is gonna be three, four; three, five; three, nine; eight, four; eight, five;
and eight, nine. So now, what we’re going to do is
find the union between these two sets. Union is this shape here we’ve seen
as a set notation of a U. And the union of two sets means a
value that is an element of either A or B. So an element of the first set or
an element of the second set.
And if we look at it as a Venn
diagram, we could see that if we had sets A and B, if we wanted the union of those
two sets, it’s any value within those sets. So therefore, we can say that the
union of 𝑋 multiplied by 𝑌 and 𝑌 multiplied by 𝑍 is gonna be all the ordered
pairs in 𝑋 multiplied by 𝑌 or 𝑌 multiplied by 𝑍. So therefore, we can say that we’re
gonna include all the values that we’ve got. So for the ordered pairs, we can
say that the union of the Cartesian product of 𝑋 and 𝑌 and the Cartesian product
of 𝑌 and 𝑍 is gonna give us the set of coordinates eight, three; eight, eight;
three, four; three, five; three, nine; eight, four; eight, five; and eight,
nine.
