Video Transcript
Using the Venn diagram, find the
complement of 𝑋 minus the complement of 𝑌.
Now, when I read the question out,
I read this notation automatically. 𝑋 and then a little superscript
dash means the complement of the set 𝑋. The complement of 𝑋 is the set of
all elements which are in the universal set, so that’s all the elements we’re
interested in in this question, but aren’t in the set 𝑋. We can write this as the universal
set 𝑈 minus the set 𝑋 if we wish. The same is true for the notation
𝑌 and then a dash. It means the complement of 𝑌. So the set of all elements which
are in our universal set 𝑈 but aren’t in the set 𝑌.
Let’s use the Venn diagram to find
each of these sets individually. For 𝑋 prime then, we’re looking
for all of the elements which are inside the universal set. So they’re in the grey box. But they aren’t in set 𝑋. So they aren’t inside the pink
oval. There are five elements of this
set, the elements one and five, which are in neither 𝑋 or 𝑌, but are in the
universal set and the elements zero, three, and eight, which are in set 𝑌, but not
in set 𝑋. So the complement of 𝑋 is the set
containing the elements zero, one, three, five, and eight.
Now, we can look for the set which
is the complement of 𝑌. Remember, this is all the elements
which are in the universal set. So they are within the grey
box. But they aren’t in set 𝑌. So they’re not within the blue
oval. We can see that this set consists
of three elements, the elements one and five, which are in the universal set but are
not in either 𝑋 or 𝑌, and the element six, which is in set 𝑋 but isn’t in set
𝑌. It’s in the pink oval, but not the
blue oval.
To find the set, the complement of
𝑋 minus the complement of 𝑌, we need to take the set zero, one, three, five, eight
and then subtract the elements one, five, and six. Let’s have a look then. So the element zero is in the
complement of 𝑋. But it isn’t in the complement of
𝑌. So it will be in the complement of
𝑋 minus the complement of 𝑌. The element one is in both of these
sets, which means it isn’t in the complement of 𝑋 minus the complement of 𝑌. The element three is in the
complement of 𝑋. But it isn’t in the complement of
𝑌. So we’re not subtracting it off,
which means that the element three will be in this set.
The element five is in both sets,
which means we will be taking this element out. So we do not have it in the
complement of 𝑋 minus the complement of 𝑌. The element eight is in the
complement of 𝑋 but isn’t in the complement of 𝑌. So we keep hold of this
element. And finally, there is one element
in the complement of 𝑌, which we haven’t mentioned. It is the element six. But remember, we’re starting with
the elements which are in the complement of 𝑋. So our starting set of values is
zero, one, three, five, and eight. And then, we’re just taking out of
this set the elements that also appear in the complement of 𝑌. As the element six didn’t appear in
our starting set, it makes no difference if we try to take it out.
We’re left then with three
elements, which are in the complement of 𝑋 but aren’t in the complement of 𝑌. Those elements are zero, three, and
eight. We can also see this directly on
our Venn diagram. The elements zero, three, and eight
are elements which we underlined in orange but didn’t underline in pink. So they’re elements that are in the
complement of 𝑋 but aren’t in the complement of 𝑌.
