In this video,
we will look at three examples
of problem solving with Venn diagrams.
The numbers in the Venn diagram represent the cardinality,
or number of elements, in a particular set.
For the first example, we're asked to select the box
or boxes for each region that represents
students taking all three courses.
Notice how the three courses are History, Math, and English.
Because we're looking for the region that represents
the students taking all three courses,
we are looking for the intersection of the three sets,
which would be this middle region here.
This set represents the students taking all three courses,
and therefore we check this box here.
Notice how there are six students taking all three courses.
So while we're done here, let's review some notation.
The number of elements in a set
is called the cardinality of the set,
indicated using either of these two notations here.
And we have a symbol for union,
and we have a symbol for intersection.
Because we are looking for the intersection
of the three sets,
if we call the three sets set M, set H, and set E,
we can state the cardinality
of set M intersect set H intersect set E is equal to six.
Let's look at a second example.
Select the box or boxes for the region that represents
students taking History or Math and not English.
Let's do this in two steps.
Let's first find the region that represents
students taking History or Math,
which would be the union of set H and set M.
Or means union.
Let's indicate this in blue.
The students taking History or Math
would need to be in H union M, which is this set here.
This is the first step, not our final answer.
And then we have "and not English," so from this set,
we must exclude the students that are taking English,
which would be these students here.
So we have the union and then we exclude this set here,
leaving us with a final set in red.
This set represents these students taking History or Math
and not English.
So we check this box, this box, and this box.
Notice how we have a total of seven students.
Again, this would be our notation.
We have the cardinality of first History or Math
would be set H union set M.
So H union M.
And then we have "and not English."
And means intersection.
Not English is the complement of English,
in this case, the complement of set E,
so we have the intersection of E complement.
Let's indicate E complement with set E
with a bar over the E.
Another way to express E complement
would be something like this.
We'll go ahead and use E bar.
This is equal to four plus zero plus three,
which is equal to seven.
Let's take a look at one more example.
Select the box or boxes for each region that represents
students taking English and History, but not Math.
So again, we'll do this in two steps.
Let's first find the set of students taking English
and History, which would be the intersection
of set E and set H.
In blue, that would be this set here.
These are the students taking English and History.
But then it says, "but not Math,"
which would be the same as And not Math,
so from this set, we exclude the students taking Math,
which would be these six students here,
and our final set is this set here.
This set represents the students,
or in this case, the student,
taking English and History, but not Math.
So we check this box here.
And again, this would be our notation.
We have the cardinality of, first,
History and English would be set E intersect set H.
"But not Math" is the same as And not Math,
so we have the intersection of Math complement,
which would be the complement of set M,
and this is equal to one.
I hope you found this helpful.