The Venn diagram
shows a cardinality of the sets.
Let's determine the following cardinalities.
First, we have the cardinality of set A,
which is equal to the number of elements in set A.
So here we have set A.
And again, the numbers indicate the cardinality in each set,
and therefore the cardinality of set A
is equal to one plus 12 plus two plus 11,
which is equal to 26.
The cardinality of set A is equal to 26.
Next, we have the cardinality of C complement,
where C complement is the set of elements not in set C
or outside set C.
So if this is set C,
C complement is the set of elements not in set C,
which is this set here.
The number of elements in C complement
or the number of elements not in C
is equal to one plus 11 plus nine plus 10,
which is equal to 31.
The cardinality of C complement is equal to 31.
Next, we have the cardinality of A intersect C,
where the intersection of two sets,
A and B contains only the elements
that are in both sets A and B,
so we're looking for the number of elements
in set A and in set C.
A intersect C is this set here,
the set of elements that are in set C and in set A.
And therefore the cardinality of A intersect C
is 12 plus two,
which is 14.
Next, we have the cardinality of A union B,
where the union of two sets, A and B,
contains all the elements in either set A or set B,
or in both sets.
The set A union B is this set here.
This is a set that contains the elements
that are in set A, or set B, or in both.
And therefore the cardinality of A union B
is equal to one plus 12 plus two plus 11
plus nine plus seven,
which is equal to 42.
The cardinality of A union B is equal to 42.
Next, we have the cardinality of A union B union C,
which is equal to the number of elements
that are in A or B or C ,
or two or three of the sets.
So the set A union B union C
is this set here.
This is the set that contains the elements in set A,
or set B, or set C,
and therefore the cardinality of A union B union C
is equal to one plus nine plus eight,
plus 11 plus seven plus 12,
plus two,
which is equal to 50.
So the last example,
we have the cardinality of A intersect B intersect C,
which is equal to the number of elements in set A,
and set B, and set C.
A intersect B intersect C
is this set here.
This is a set of elements that are in set A,
and set B, and set C.
And therefore the cardinality of A intersect B intersect C
is equal to two.
I hope you found this helpful.