Suppose the cardinality of set A equals 18
and the cardinality of set B equals nine.
What is the cardinality of A union B
plus the cardinality of A intersect B?
What we learned earlier
for any two finite sets A and B
the cardinality of A union B
equals the cardinality of A
plus the cardinality of B
minus the cardinality of A union B.
And since we're looking for the cardinality of A union B
plus the cardinality of A intersect B.
Notice here if we add
the cardinality of A intersect B
to both sides of the equation,
we get the equation
the cardinality of A union B
plus the cardinality of A intersect B
equals the cardinality of A plus the cardinality of B.
So again, to answer the question,
the cardinality of A union B
plus the cardinality of A intersect B
equals the cardinality of A plus the cardinality of B.
And we are given the cardinality of A is 18.
And the cardinality of B is nine,
which gives us 18 plus nine, which is equal to 27.
So while this is the answer to the question,
let's verify this using the three cases
where one case is, A and B are disjoint.
Next, where B is a subset of A,
and then the third case
where A and B are not disjoint.
So when A and B are disjoint,
let's go ahead and label the cardinalities.
The cardinality of A is 18.
The cardinality of B is nine.
And then when B is a subset of A,
we know the cardinality of B is nine.
And since the cardinality of A is 18,
there are nine elements in A that are not in B.
Meaning nine elements in this region here.
Notice there are 18 elements in set A.
And then finally, when A and B are not disjoint.
Let's just assume there are three elements
in the intersection of A and B here.
Since the cardinality of A is 18,
that means there are 15 elements only in A.
And then because the cardinality of B is nine,
there are six elements only in B.
And again, let's go ahead
and just verify this formula is correct.
Which we know it should be
because we already know
the original formula for the cardinality
of A union B is correct.
So for case one,
the cardinality of A union B
is equal to a number of elements in A or B,
which you can see when they are disjoint
is just going to be 18 plus nine, or 27.
And then plus the cardinality of A intersect B
is equal to a number of elements in A and in B,
which when A and B are disjoint, is equal to zero.
And of course we know the cardinality of A
plus the cardinality of B is 18 plus nine.
And as we do get 27 equals 27.
And now for case two, when B is a subset of A,
the cardinality of A union B
is equal to a number of elements in A or in B.
But because all the elements
in B are also elements in A,
the set A union B is just the set A,
which has a cardinality of 18.
The cardinality of A union B is 18.
Plus the cardinality of A intersect B
is the number of elements in A and in B.
And again, because B is a subset of A,
A intersect B is just equal to B.
All the elements in B
are the elements that are in A and in B.
And therefore the cardinality of A intersect B
equals the cardinality of B, which is nine.
And of course the cardinality of A
plus the cardinality of B
we know is 18 plus nine.
And we do get 27 equals 27.
And now let's look at case three.
The cardinality of A union B
is equal to the number of elements in A or in B,
which is this set here.
Notice how we have 15 plus three plus six,
which is equal to 24.
Plus the cardinality of A intersect B
is equal to the number of elements in A and in B,
which is this set here.
Notice the cardinality is three.
And of course, we know the right side is 18 plus nine.
And of course we do get 27 equals 27.
I hope you found this helpful.