We're asked to find
the cardinality of each set.
The cardinality of a set is a number
of distinct elements in the set.
So looking at set A, the cardinality is equal
to the number of elements in the set.
So we could count them one by one,
but let's look at a formula we can use,
when we have a set that consists
of an ordered sequential list of integers.
The number of integers in an ordered sequential list
is equal to the greatest value, minus the least value
plus one.
Let's look at a simple example.
Let's take a look at a simple example of why this works.
Let's say we have set E that contains the elements,
one, two, three, four and five.
We can easily see set E contains five elements
and therefore the cardinality of set E is equal to five.
But using the formula, another way to determine
the number of elements in the set,
because we have an ordered sequential list of integers,
is to take the largest value of five,
subtract the least value of one, and then add one.
Notice this gives us four plus one, which is equal to five.
Just keep in mind this only works when we have a list
of integers that are ordered and sequential,
meaning no integers are missing from the list.
So now going back to set A, if we don't want to
count the number of elements in the set one by one,
we can use the formula, the greatest value of 25
minus the least value of eight plus one
will give us number of elements in the set.
Well 25 minus eight is equal to 17.
17 plus one is equal to 18.
There are 18 elements in set A, and therefore
the cardinality of set A is 18.
I guess another way we could think of this is,
the number of elements from 11 to 25 is 15,
and then we have three more, eight, nine, and 10,
15 plus three is 18.
Whichever way, the Cardinal is 18.
For set B, set B is a set of all integers X,
such that X is greater than equal to negative five
and less than equal to 18.
Let's go ahead and show set B as a list.
The least element in set B is negative five,
and then we have negative four, negative three,
negative two and so on.
The last three elements in the set would be
16, 17, and 18.
So again, the cardinality is number of elements in set B.
Well, the number of elements from one to 18 is 18.
The number of elements from negative five to zero is six,
18 plus six is 24.
The cardinality of set B is 24.
Or again if we want to use the formula,
we would have 18 minus negative five, plus one,
well 18 minus negative five simplifies 18 plus five or 23,
23 plus one is equal to 24.
For the last example we're asked to determine
the cardinality of the intersection of C and D.
Let's write set C and D as lists.
So set C, is a set of all natural numbers X,
such that X as less or equal to 20.
And therefore C contains the elements zero, one, two, three
all the way out to 20.
I'm going to go and list all of them.
And that's right, set D is a list, set D is a set
of all natural numbers X, such that X is prime.
Remember a prime number is greater than one
and only has two factors one and itself.
So the first element in D is two, followed by three
and then five, seven, 11, 13, 17,
19, 23, and dot, dot, dot.
We can go ahead and stop here, 'cause remember
we're looking for the intersection of C and D
and the greatest element in C is 20.
So now let's determine the intersection of C and D.
So they begin to determine the cardinality
of the intersection of C and D.
C intersect D is a set containing the elements
that are in C and in D.
Which notice are going to be the prime numbers
from two through 19.
Notice the elements of two, three, five, seven,
11, 13, 17 and 19, are in C and in D.
The number of elements in the intersection of C and D,
is one, two, three, four, five, six, seven, eight.
The cardinality of C intersect D equals eight.
I hope you found this helpful.