In this example, we're given the universal set contains the whole numbers
one through twenty, where A, B and C are subsets containing the elements given here.
We want to find the number of elements in the set A intersect C, intersect B complement.
The number of elements in this set is also called the cardinality of the set.
Let's solve this problem using a Venn diagram. So to begin, let's focus on the
innermost part of the Venn diagram, which would be this region here, the intersection of A, B, and C.
The elements in here would have to be in A, and B, and C.
So looking at our three sets, let's see if we can determine any elements that are in all three sets.
Notice how eight, nine and fifteen are the only elements that are in all three sets,
therefore, these three elements would be in this region here.
Again we have eight, nine, and fifteen.
Now let's focus on this region here.
This region contains the elements that are only in A and B.
Looking at sets A and B, let's determine the elements that are only in A and B.
Notice the element only in A and B is ten.
So ten goes in this region here.
Next let's focus on this region here. This would contain the elements that are
only in B and C. So looking at B and C, notice four, six, and fourteen are only in
B and C, and therefore, these three elements go in this region here.
Next let's focus on this region here.
This contains the elements that are only in A and C. So looking at only sets A and C,
notice that eleven, and thirteen are only in A and C,
so eleven and thirteen go in this region here. Now if we go back to sets A, B, and C,
we can determine the elements that are only in A, only in B, and only in C.
Looking at set A, notice one is only in A.
Looking at B, notice seven is only in B.
Looking at set C, notice three, and twelve are only in C.
And we're not quite done yet notice how the universal set contains the whole numbers
one through twenty. And notice how the number two is not given in sets A, B, or C.
So two would be in the universal set here.
And then five is also not in A, B, or C, so five is in the universal set.
And then we have six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen are all here.
But sixteen through twenty are not in sets A, B, or C,
so they would also be in the universal set. So sixteen, seventeen, eighteen, nineteen, and twenty.
And now we'll find the set A intersect C, intersect B compliment, then determine how many elements are
in that set. So we'll first find A intersect C,
which looking at the Venn diagram would be where A and C intersect, or where they overlap.
Take a look at sets A and C here. Notice how this would be the region here,
containing the elements that are in A, and also in C.
So this is A intersect C,
and now intersect B compliment.
Remember B complement would be all the elements that are not in B.
All the elements not in B would be all the elements that are not in this region here.
So they would be in this region.
Notice these elements here in the universal set are not in B, so we also
have to include them.
This orange region contains all the elements that are not in B.
So we're looking for the intersection of these two sets.
So we're looking for elements that are in both sets. Notice how that would be this
region here containing the elements eleven and twelve, and therefore, the number of elements
in A intersect C, intersect B complement is two.
We'll take a look at this problem again,
and see if we determine the number of elements without using a Venn diagram.
I hope you found this helpful.