Video Transcript
Determine the union of the
Cartesian products 𝑋 and 𝑌 and 𝑌 and 𝑍 using the Venn diagram below.
We recall that the union of two
sets is those elements that occur in the first set or the second set. The Cartesian product of any two
sets is the set of all ordered pairs. We can see from the Venn diagram
that set 𝑋 contains the element one; set 𝑌 contains the elements seven and five;
set 𝑍 contains the elements two, zero, and five. Note that the number five appears
in both set 𝑌 and set 𝑍 as it is the intersection of these two sets.
We can use the elements of set 𝑋
and set 𝑌 to work out the Cartesian product of 𝑋 and 𝑌. This contains the two ordered pairs
one, seven and one, five. Note that the 𝑋-values are the
first numbers in our ordered pair and the 𝑌-values are our second numbers. We can repeat this process for the
Cartesian product of set 𝑌 and set 𝑍. In this Cartesian product, we have
six ordered pairs: seven, two; seven, zero; seven, five; five, two; five, zero; and
five, five.
We now need to work out the union
of these two sets. This will include all the ordered
pairs that are either in the Cartesian product of 𝑋 and 𝑌 or in the Cartesian
product of 𝑌 and 𝑍. None of the ordered pairs are
repeated. Therefore, we need to include all
eight ordered pairs. The correct answer is the eight
ordered pairs one, seven; one, five; seven, two; seven, zero; seven, five; five,
two; five, zero; and five, five.
