We are told to let the set S be the universal set,
where S contains integers from one through 20.
And then let sets A and B be subsets of S,
where set A contains the elements
one, three, eight, nine, 11, 12, 13, 17,
and set B contains the elements
two, three, five, six, seven, eight, 10, 14, 16, 17, 18, 20.
We are asked to determine the cardinality
of the following sets.
A universal set is a set that contains
all the elements we are interested in
or the ones we are considering.
And the number of elements in a set
is the cardinality of that set.
The cardinality of set A is often noted
using one of these notations.
So, first we have the cardinality of set A.
Looking at set A, notice how set A contains
one, two, three, four, five, six, seven, eight elements,
and therefore, the cardinality of set A is equal to eight.
Next, we have the cardinality of A complement,
where the complement of set A contains
everything that is not in the set A.
So the cardinality of A complement
is equal to the number of elements
that are in the universal set, but not in set A.
And again, we know that set A contains eight elements.
And we know the universal set contains 20 elements.
So we can also say the cardinality of A complement
is equal to the cardinality of the universal set S
minus the cardinality of set A.
If we take the number of elements in the universal set
and subtract the number of elements in set A,
this will leave us with the number of elements
in A complement or the number of elements
that are not in set A.
So the cardinality of set S, universal set, is 20.
The cardinality of set A we know is eight.
And 20 minus eight is equal to 12.
There are 12 elements in the universal set
that are not in set A.
The cardinality of A complement is equal to 12.
Next, we have the cardinality of set B.
So looking at set B, there are one, two, three, four, five,
six, seven, eight, nine, 10, 11, 12 elements in set B.
The cardinality of set B is equal to 12.
Next, we have the cardinality of A intersect B,
where the intersection of two sets A and B
contains only the elements that are in both sets A and B.
Let's begin by determining the set A intersect B
by determining which elements are in both set A and set B.
So looking at the two sets, notice how both sets
contain the element three.
Both sets contain the element eight.
And both sets contain the element 17.
There are no other elements that are in both sets,
and therefore, A intersect B is the set containing
three, eight, and 17.
Because there are three elements in A intersect B,
the cardinality of A intersect B is equal to three.
Next, we have the cardinality of A union B,
where the union of sets A and B contains
all the elements in either set A or set B.
Let's begin by determining the set A union B.
This will contain all the elements
that are in either A or B.
And let's go in order from least to greatest.
One is in set A, one is in the union.
Two is in set B, two is in the union.
Three is in both, three is in the union.
Next, five, six, and seven are in set B.
Five, six, and seven are in the union.
Eight is in both, eight is in the union.
Nine is in set A, nine is in the union.
10 is in B, 10 is in the union.
11, 12, and 13 are in set A, which are also in the union.
14 and 16 are in set B, which are in the union.
17 is in both, 17 is in the union.
And then we have 18 and 20 in set B,
which are also in the union.
So here we have A union B.
Let's count the number of elements in this set.
We have one, two, three, four, five, six,
seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17.
Because there are 17 elements in A union B,
the cardinality of A union B is equal to 17.
I hope you found this helpful.